Stochastic spiking network learning apparatus and methods

ABSTRACT

Generalized learning rules may be implemented. A framework may be used to enable adaptive spiking neuron signal processing system to flexibly combine different learning rules (supervised, unsupervised, reinforcement learning) with different methods (online or batch learning). The generalized learning framework may employ time-averaged performance function as the learning measure thereby enabling modular architecture where learning tasks are separated from control tasks, so that changes in one of the modules do not necessitate changes within the other. Separation of learning tasks from the control tasks implementations may allow dynamic reconfiguration of the learning block in response to a task change or learning method change in real time. The generalized spiking neuron learning apparatus may be capable of implementing several learning rules concurrently based on the desired control application and without requiring users to explicitly identify the required learning rule composition for that task.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to a co-owned and co-pending U.S. patentapplication Ser. No. 13/XXX,XXX entitled “STOCHASTIC APPARATUS ANDMETHODS FOR IMPLEMENTING GENERALIZED LEARNING RULES”, [attorney docket021672-0405921, client reference BC201202A] filed contemporaneouslyherewith, co-owned U.S. patent application Ser. No. 13/XXX,XXX entitled“IMPROVED LEARNING STOCHASTIC APPARATUS AND METHODS”, [attorney docket021672-0407763, client reference BC201208A], filed contemporaneouslyherewith, and co-owned U.S. patent application Ser. No. 13/XXX,XXXentitled “DYNAMICALLY RECONFIGURABLE STOCHASTIC SPIKING NETWORKAPPARATUS AND METHODS”, [attorney docket 021672-0407729, clientreference BC201211A], filed contemporaneously herewith, each of theforegoing incorporated herein by reference in its entirety.

COPYRIGHT

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BACKGROUND

1. Field of the Disclosure

The present disclosure relates to implementing generalized learningrules in stochastic spiking neuron systems.

2. Description of Related Art

Adaptive signal processing systems are well known in the arts ofcomputerized control and information processing. One typicalconfiguration of an adaptive system of prior art is shown in FIG. 1. Thesystem 100 may be capable of changing or “learning” its internalparameters based on the input 102, output 104 signals, and/or anexternal influence 106. The system 100 may be commonly described using afunction 110 that depends (including probabilistic dependence) on thehistory of inputs and outputs of the system and/or on some externalsignal r that is related to the inputs and outputs. The functionF(x,y,r) may be referred to as a “performance function”. The purpose ofadaptation (or learning) may be to optimize the input-outputtransformation according to some criteria, where learning is describedas minimization of an average value of the performance function F.

Although there are numerous models of adaptive systems, these typicallyimplement a specific set of learning rules (e.g., supervised,unsupervised, reinforcement). Supervised learning may be the machinelearning task of inferring a function from supervised (labeled) trainingdata. Reinforcement learning may refer to an area of machine learningconcerned with how an agent ought to take actions in an environment soas to maximize some notion of reward (e.g., immediate or cumulative).Unsupervised learning may refer to the problem of trying to find hiddenstructure in unlabeled data. Because the examples given to the learnerare unlabeled, there is no external signal to evaluate a potentialsolution.

When the task changes, the learning rules (typically effected byadjusting the control parameters w={w_(i), w₂, . . . , w_(n)}) may needto be modified to suit the new task. Hereinafter, the boldface variablesand symbols with arrow superscripts denote vector quantities, unlessspecified otherwise. Complex control applications, such as for example,autonomous robot navigation, robotic object manipulation, and/or otherapplications may require simultaneous implementation of a broad range oflearning tasks. Such tasks may include visual recognition ofsurroundings, motion control, object (face) recognition, objectmanipulation, and/or other tasks. In order to handle these taskssimultaneously, existing implementations may rely on a partitioningapproach, where individual tasks are implemented using separatecontrollers, each implementing its own learning rule (e.g., supervised,unsupervised, reinforcement).

One conventional implementation of a multi-task learning controller isillustrated in FIG. 1A. The apparatus 120 comprises several blocks 120,124, 130, each implementing a set of learning rules tailored for theparticular task (e.g., motor control, visual recognition, objectclassification and manipulation, respectively). Some of the blocks(e.g., the signal processing block 130 in FIG. 1A) may further comprisesub-blocks (e.g., the blocks 132, 134) targeted at different learningtasks. Implementation of the apparatus 120 may have several shortcomingsstemming from each block having a task specific implementation oflearning rules. By way of example, a recognition task may be implementedusing supervised learning while object manipulator tasks may comprisereinforcement learning. Furthermore, a single task may require use ofmore than one rule (e.g., signal processing task for block 130 in FIG.1A) thereby necessitating use of two separate sub-blocks (e.g., blocks132, 134) each implementing different learning rule (e.g., unsupervisedlearning and supervised learning, respectively).

Artificial neural networks may be used to solve some of the describedproblems. An artificial neural network (ANN) may include a mathematicaland/or computational model inspired by the structure and/or functionalaspects of biological neural networks. A neural network comprises agroup of artificial neurons (units) that are interconnected by synapticconnections. Typically, an ANN is an adaptive system that is configuredto change its structure (e.g., the connection configuration and/orneuronal states) based on external or internal information that flowsthrough the network during the learning phase.

A spiking neuronal network (SNN) may be a special class of ANN, whereneurons communicate by sequences of spikes. SNN may offer improvedperformance over conventional technologies in areas which includemachine vision, pattern detection and pattern recognition, signalfiltering, data segmentation, data compression, data mining, systemidentification and control, optimization and scheduling, and/or complexmapping. Spike generation mechanism may be a discontinuous process(e.g., as illustrated by the input spikes sx(t) 220, 222, 224, 226, 228,and output spikes sy(t) 230, 232, 234 in FIG. 2) and a classicalderivative of function F(s(t)) with respect to spike trains sx(t), sy(t)is not defined.

Even when a neural network is used as the computational engine for theselearning tasks, individual tasks may be performed by a separate networkpartition that implements a task-specific set of learning rules (e.g.,adaptive control, classification, recognition, prediction rules, and/orother rules). Unused portions of individual partitions (e.g., motorcontrol when the robotic device is stationary) may remain unavailable toother partitions of the network that may require increased processingresources (e.g., when the stationary robot is performing facerecognition tasks). Furthermore, when the learning tasks change duringsystem operation, such partitioning may prevent dynamic retargeting(e.g., of the motor control task to visual recognition task) of thenetwork partitions. Such solutions may lead to expensive and/orover-designed networks, in particular when individual portions aredesigned using the “worst possible case scenario” approach. Similarly,partitions designed using a limited resource pool configured to handlean average task load may be unable to handle infrequently occurring highcomputational loads that are beyond a performance capability of theparticular partition, even when other portions of the networks havespare capacity.

By way of illustration, consider a mobile robot controlled by a neuralnetwork, where the task of the robot is to move in an unknownenvironment and collect certain resources by the way of trial and error.This can be formulated as reinforcement learning tasks, where thenetwork is supposed to maximize the reward signals (e.g., amount of thecollected resource). While in general the environment is unknown, theremay be possible situations when the human operator can show to thenetwork desired control signal (e.g., for avoiding obstacles) during theongoing reinforcement learning. This may be formulated as a supervisedlearning task. Some existing learning rules for the supervised learningmay rely on the gradient of the performance function. The gradient forreinforcement learning part may be implemented through the use of theadaptive critic; the gradient for supervised learning may be implementedby taking a difference between the supervisor signal and the actualoutput of the controller. Introduction of the critic may be unnecessaryfor solving reinforcement learning tasks, because direct gradient-basedreinforcement learning may be used instead. Additional analyticderivation of the learning rules may be needed when the loss functionbetween supervised and actual output signal is redefined.

While different types of learning may be formalized as a minimization ofthe performance function F, an optimal minimization solution oftencannot be found analytically, particularly when relationships betweenthe system's behavior and the performance function are complex. By wayof example, nonlinear regression applications generally may not haveanalytical solutions. Likewise, in motor control applications, it maynot be feasible to analytically determine the reward arising fromexternal environment of the robot, as the reward typically may bedependent on the current motor control command and state of theenvironment.

Moreover, analytic determination of a performance function F derivativemay require additional operations (often performed manually) forindividual new formulated tasks that are not suitable for dynamicswitching and reconfiguration of the tasks described before.

Some of the existing approaches of taking a derivative of a performancefunction without analytic calculations may include a “brute force”finite difference estimator of the gradient. However, these estimatorsmay be impractical for use with large spiking networks comprising many(typically in excess of hundreds) parameters.

Derivative-free methods, specifically Score Function (SF), also known asLikelihood Ratio (LR) method, exist. In order to determine a directionof the steepest descent, these methods may sample the value of F(x,y) indifferent points of parameter space according to some probabilitydistribution. Instead of calculating the derivative of the performancefunction F(x,y), the SR and LR methods utilize a derivative of thesampling probability distribution. This process can be considered as anexploration of the parameter space.

Although some adaptive controller implementations may describereward-modulated unsupervised learning algorithms, these implementationsof unsupervised learning algorithms may be multiplicatively modulated byreinforcement learning signal and, therefore, may require the presenceof reinforcement signal for proper operation.

Many presently available implementations of stochastic adaptiveapparatuses may be incapable of learning to perform unsupervised taskswhile being influenced by additive reinforcement (and vice versa). Manypresently available adaptive implementations may be task-specific andimplement one particular learning rule (e.g., classifier unsupervisedlearning), and such devices invariably require retargeting (e.g.,reprogramming) in order to implement different learning rules.Furthermore, presently available methodologies may not be capable ofimplementing generalized learning, where a combination of differentlearning rules (e.g., reinforcement, supervised and supervised) are usedsimultaneously for the same application (e.g., platform motionstabilization), thereby enabling, for example, faster learningconvergence, better response to sudden changes, and/or improved overallstability, particularly in the presence of noise.

Stochastic Spiking Neuron Models

Where certain elements of these implementations can be partially orfully implemented using known components, only those portions of suchknown components that are necessary for an understanding of the presentdisclosure will be described, and detailed descriptions of otherportions of such known components will be omitted so as not to obscurethe disclosure.

Learning rules used with spiking neuron networks may be typicallyexpressed in terms of original spike trains instead of their secondaryfeatures (e.g., the rate or the latency from the last spike). The resultis that a spiking neuron operates on spike train space, transforming avector of spike trains (input spike trains) into single element of thatspace (output train). Dealing with spike trains directly may be achallenging task. Not every spike train can be transformed to anotherspike train in a continuous manner. One common approach is to describethe task in terms of optimization of some function and then use gradientapproaches in the parameter space of the spiking neuron. Howevergradient methods on discontinuous spaces such as spike trains space arenot well developed. One approach may involve smoothing the spike trainsfirst. Here output spike trains are smoothed with introduction ofprobabilistic measure on a spike trains space. Describing the spikepattern from a probabilistic point of view may lead to fruitfulconnections with the huge amount of topics within information theory,machine learning, Bayesian inference, statistical data analysis etc.This approach makes spiking neurons a good candidate to use SF/LRlearning methods.

One technique frequently used when constructing learning rules in aspiking network, comprises application of a random exploration processto a spike generation mechanism of a spiking neuron. This is oftenimplemented by introducing a noisy threshold: probability of a spikegeneration may depend on the difference between neuron's membranevoltage and a threshold value. The usage of probabilistic spiking neuronmodels, in order to obtain gradient of the log-likelihood of a spiketrain with respect to neuron's weights, may comprise an extension ofHebbian learning framework to spiking neurons. The use of thelog-likelihood gradient of a spike train may be extended to supervisedlearning. In some approaches, information theory framework may beapplied to spiking neurons, as for example, when deriving optimallearning rules for unsupervised learning tasks via informational entropyminimization.

An application of the OLPOMDM algorithm to the solution of thereinforcement learning problems with simplified spiking neurons has beendone. Extending of this algorithm to more plausible neuron model hasbeen done. However no generalizations of the OLPOMDM algorithm have beendone in order to use it unsupervised and supervised learning in spikingneurons. An application of reinforcement learning ideas to supervisedlearning has been described, however only heuristic algorithms withoutconvergence guarantees have been used.

For a neuron, the probability of an output spike train, y, to havespikes at times t_f with no spikes at the other times on a time interval[0, T], given the input spikes, x, may be given by the conditionalprobability density function p(y|x) as:

$\begin{matrix}{p\left( {{y\left. x \right)} = {\Pi_{t_{f}}{\lambda \left( t_{f} \right)}^{{- f_{0}^{T}}{\lambda {(\tau)}}{\tau}}}} \right.} & \left( {{Eqn}.\mspace{11mu} 1} \right)\end{matrix}$

where λ(t) represents an instantaneous probability density (“hazard”) offiring.

The instantaneous probability density of the neuron can depend onneuron's state q(t): λ(t)≡λ(q(t)). For example, it can be definedaccording to its membrane voltage u(t) for continuous time chosen as anexponential stochastic threshold:

λ(t)=λ_(o) e ^(κ(u(t)-θ))  (Eqn. 2)

where u(t) is the membrane voltage of the neuron, θ is the voltagethreshold for generating a spike, x is the probabilistic parameter, andλ₀ is the basic (spontaneous) firing rate of the neuron.

Some approaches utilize sigmoidal stochastic threshold, expressed as:

$\begin{matrix}{{\lambda (t)} = \frac{\lambda_{0}}{1 - ^{- {\kappa {({{u{(t)}} - \theta})}}}}} & \left( {{Eqn}.\mspace{11mu} 3} \right)\end{matrix}$

or an exponential-linear stochastic threshold:

λ(t)=λ₀ ln(1+e ^(κ(u(t)-θ)))  (Eqn. 4)

where λ₀, κ, θ are parameters with a similar meaning to the parametersin the exponential threshold model Eqn. 2.

Models of the stochastic threshold exist comprising refractorymechanisms that modulate the instantaneous probability of firing afterthe last output spike λ(t)={circumflex over (λ)}(t)R(t,t_(last) ^(out)),where {circumflex over (λ)}(t) is the original stochastic thresholdfunction (such as exponential or other), and R(t_(last) ^(out)−t) is thedynamic refractory coefficient that depends on the time since the lastoutput spike t_(last) ^(out).

For discrete time steps, an approximation for the probabilityΛ(u(t))ε(0,1] of firing in the current time step may be given by:

Λ(u(t))=1−e ^(−λ(u(t))Δt)  (Eqn. 5)

where Δt is time step length.

In one dimensional deterministic spiking models, such asIntegrate-and-Fire (IF), Quadratic Integrate-and-Fire (QIF) and others,membrane voltage u(t) is the only one state variable (q(t)≡u(t)) that is“responsible” for spike generation through deterministic thresholdmechanism. There also exist plenty of more complex multidimensionalspiking models. For example, a simple spiking model may comprise twostate variables where only one of them is compared with a thresholdvalue. However, even detailed neuron models may be parameterized using asingle variable (e.g., an equivalent of “membrane voltage” of biologicalneuron) and use it with a suitable threshold in order to determine thepresence of spike. Such models are often extended to describe stochasticneurons by replacing deterministic threshold with a stochasticthreshold.

Generalized dynamics equations for spiking neurons models are oftenexpressed as a superposition of input, interaction between the inputcurrent and the neuronal state variables, and neuron reset after thespike as follows:

$\begin{matrix}{\frac{\overset{\rightarrow}{q}}{t} = {{V\left( \overset{\rightarrow}{q} \right)} + {\overset{\;}{\sum_{t^{out}}}{{R\left( \overset{\rightarrow}{q} \right)}{\delta \left( {t - t^{out}} \right)}}} + {{G\left( \overset{\rightarrow}{q} \right)}I^{ext}}}} & \left( {{Eqn}.\mspace{11mu} 6} \right)\end{matrix}$

where:

is a vector of internal state variables (e.g., comprising membranevoltage); I^(ext) is external input to the neuron; F—is the functionthat defines evolution of the state variables; G describes theinteraction between the input current and the state variables (forexample, to model synaptic depletion); and R describes resetting thestate variables after the output spikes at t^(out).

For example, for IF model the state vector and the state model may beexpressed as:

{right arrow over (q)}≡u(t);V({right arrow over (q)})=−Cu;R({right arrowover (q)})=u ^(res) −u;G({right arrow over (q)})=1,  (Eqn. 7)

where C is a membrane constant, and u_(res) is the value to whichvoltage is set after output spike (reset value). Accordingly, Eqn. 6becomes:

$\begin{matrix}{\frac{u}{t} = {{- {Cu}} + {\sum\limits_{t^{out}}{\left( {u_{refr} - u} \right){\delta \left( {t - t^{out}} \right)}}} + I^{ext}}} & \left( {{Eqn}.\mspace{11mu} 8} \right)\end{matrix}$

For some simple neuron models, Eqn. 6 may be expressed as:

$\begin{matrix}{{\frac{v}{t} = {{0.04v^{2}} + {5v} + 140 - u + {\sum\limits_{t^{out}}{\left( {c - v} \right){\delta \left( {t - t^{out}} \right)}}} + I^{ext}}}\mspace{20mu} {\frac{u}{t} = {{a\left( {{bv} - u} \right)} + {d{\sum\limits_{t^{out}}{\delta \left( {t - t^{out}} \right)}}}}}\mspace{20mu} {where}} & \left( {{Eqn}.\mspace{11mu} 9} \right) \\{\mspace{79mu} {{{\overset{\rightarrow}{q} = \begin{pmatrix}{v(t)} \\{u(t)}\end{pmatrix}};{{V\left( \overset{\rightarrow}{q} \right)} = \begin{pmatrix}{{0.04{v^{2}(t)}} - {5{v(t)}} + 140 - {u(t)}} \\{a\left( {{{bv}(t)} - {u(t)}} \right)}\end{pmatrix}};}\mspace{20mu} {{{{R\left( \overset{\rightarrow}{q} \right)} = \begin{pmatrix}{c - {v(t)}} \\d\end{pmatrix}};{{G\left( \overset{\rightarrow}{q} \right)} = \begin{pmatrix}1 \\0\end{pmatrix}}},}}} & \left( {{Eqn}.\mspace{11mu} 10} \right)\end{matrix}$

and a, b, c, d are parameters of the model.

Many presently available implementations of stochastic adaptiveapparatuses may be incapable of learning to perform unsupervised taskswhile being influenced by additive reinforcement (and vice versa). Manypresently available adaptive implementations may be task-specific andimplement one particular learning rule (e.g., classifier unsupervisedlearning), and such devices invariably require retargeting (e.g.,reprogrammed) in order to implement different learning rules.

Accordingly, there is a salient need for machine learning apparatus andmethods to implement generalized stochastic learning in spiking networksthat is configured to handle simultaneously any learning rulecombination (e.g., reinforcement, supervised, unsupervised, online,batch) and is capable of, inter alia, dynamic reconfiguration using thesame set of network resources.

SUMMARY

The present disclosure satisfies the foregoing needs by providing, interalia, apparatus and methods for implementing generalized probabilisticlearning configured to handle simultaneously various learning rulecombinations.

One aspect of the disclosure relates to one or more systems and/orcomputer-implemented methods for effectuating a spiking networkstochastic signal processing system configured to implementtask-specific learning. In one implementation, the system may comprise acontroller apparatus configured to generate output control signal ybased at least in part on input signal x, the controller apparatuscharacterized by a controller state parameter S, and a control parameterw; and a learning apparatus configured to: generate an adjustment signaldw based at least in part on the input signal x, the controller stateparameter S, and the output signal y; and provide the adjustment signaldw to the controller apparatus, thereby effecting the learning where thecontrol parameter may be configured in accordance with the task; and theadjustment signal dw may be configured to modify the control parameterbased at least in part on the input signal x and the output signal y.

In some implementations, the output control signal y may comprise aspike train configured based at least in part the adjustment signal dw;and the learning apparatus may comprise a task-specific block,configured independent from the controller state parameter, thetask-specific block configured to implement the task-specific learning;and a controller-specific block, configured independent from thetask-specific learning; and the task-specific learning may becharacterized by a performance function, the performance functionconfigured to effect at least unsupervised learning rule.

In some implementations, the system may further comprise a teachinginterface operably coupled to the learning apparatus and configured toprovide a teaching signal; the teaching signal may comprise a desiredcontroller output signal; and the performance function may be furtherconfigured to effect a supervised learning rule, based at least in parton the desired controller output signal; and the teaching signal mayfurther comprise a reinforcement spike train associated with currentperformance of the controller apparatus relative to desired performance;and the performance function maybe further configured to effect areinforcement learning rule, based at least in part on the reinforcementspike train.

In some implementations, the current performance may be based at leastin part on adjustment of the control parameter from a prior state w0 tocurrent state wc; the reinforcement may be positive when the currentperformance may be closer to desired performance of the controller; andthe reinforcement may be negative when the current performance may befarther from the desired performance; and the task-specific learning maycomprise a hybrid learning rule comprising a combination of thereinforcement, the supervised and the unsupervised learning rulessimultaneous with one another.

In some implementations, the adjustment signal dw may be determined as aproduct of controller performance function F with a gradient ofper-stimulus entropy parameter h, the gradient may be determined withrespect to the control parameter w; and per-stimulus entropy parameter hmay be configured to characterize dependence of the output signal y on(i) the input signal x; and (ii) the control parameter w; and theper-stimulus entropy parameter may be determined based on a naturallogarithm of p(y|x,w), where p denotes conditional probability of signaly given signal x with respect to the control parameter w.

Another aspect of the disclosure relates to one or more apparatuses foreffectuating learning in a stochastic spiking network. In oneimplementation, a computer readable apparatus may comprise a storagemedium, the storage medium comprising a plurality of instructions toadjust a learning parameter associated with a computerized spikingneuron configured to produce output spike signal y consistent with (i)an input spike signal x, and (ii) a learning task, the instructionsconfigured to, when executed construct time derivative representation ofa trace S of a neuron state, based at least in part on the input spikesignal x and a state parameter q; obtain a realization of the trace S,based at least in part in integrating the time derivativerepresentation; and determine adjustment dw of the parameter w, based atleast in part on the trace S; and the adjustment dw may be configured totransition the neuron state towards a target state, the target stateassociated with the neuron generating the output spike signal y.

In some implementations, the integrating representation may be effectedvia symbolic integration operation, the state parameter q may beconfigured to characterize time evolution of the neuron state; therealization of the trace S may comprise an analytic solution of the timederivative representation; and the construct of the time derivativerepresentation enables to attain the integration via symbolicintegration operation.

In some implementations, the state parameter q may be configured tocharacterize time evolution of the neuron state in accordance with astate evolution process characterized by: a response mode and atransition mode, the response mode may be associated with generating aneuronal response P; state transition term V describing changes ofneuronal state in the transition mode; state transition term Rdescribing changes of state set in the response mode; and statetransition term G describing changes of state set due to the input x;the state parameter q may be configured to characterize neuron membranevoltage; and the input may comprise analog signal and the statetransition term G may be configured to describe changes of the voltagedue to the analog signal.

In some implementations, the state parameter q may comprise neuronexcitability and, the time derivative representation may comprise a sumof V, R, G each multiplicatively combined with the trace S, statetransition term V may comprise the trace S multiplicatively combinedwith a Jacobian matrix Jv configured in accordance with the transitionmode of the evolution process; state transition term R may comprise thetrace S multiplicatively combined with a Jacobian matrix Jr configuredin accordance with the response mode of the evolution process; and statetransition term G may comprise the trace S multiplicatively combinedwith a Jacobian matrix Jg configured in accordance with the input x.

In some implementations, the input may comprise feed-forward input viaan interface; and the learning parameter may comprise efficacyassociated with the interface, the interface may comprise synapticconnection and the learning parameter may comprise connection weight,the state parameter q may be configured to describe time evolution ofthe neuron state in accordance with a state evolution process,characterized by evolution process may be characterized by aninstantaneous probability density distribution IPD of generatingneuronal response P.

In one or more implementations, the instructions are further configuredto, when executed, determine derivative dλ/dw of the IPD, with respectto the learning parameter w, based at least in part on the trace S; andobtain an instantaneous score function value g, based at least in parton the derivative dλ/dw; and the determine the adjustment dw may bebased at least in part on the instantaneous score function value g,where the determination of the realization of the trace S, and thedetermination of the derivative dλ/dw, and the obtaining of theinstantaneous score function value g cooperate to produce the adjustmentdw such that a next instance of the neuron state, associated with anadjusted value w2, configured based on the current value w1 and theadjustment dw, may be closer to the target state.

In one or more implementations, computerized apparatus may be configuredto process input spike train x using hybrid learning rule, the apparatuscomprising stochastic learning block configured to produce learningsignal based at least in part on the input spike x and training signalr, the hybrid learning rule may be configured to simultaneously effectreinforcement learning rule and unsupervised learning rule.

In some implementations, the stochastic learning block may be operableaccording to a stochastic process characterized by a current state and adesired state, the process being described by at least a state variableconfigured to transition the learning block from current state to thedesired state, the training signal r may comprise a reinforcementspiking indicator associated with current performance relative todesired performance of the apparatus, the current performancecorresponding to the current state and the desired performancecorresponding to the desired state, the current performance may beeffected, at least partly, by transition from a prior state to thecurrent state, the reinforcement learning may be configured based atleast in part on the reinforcement spiking indicator so that itprovides: positive reinforcement when a distance measure between thecurrent state and the desired state may be smaller compared to thedistance measure between the prior state and the desired state; andnegative reinforcement when the distance measure between the currentstate and the desired state may be greater compared to the distancemeasure between the prior state and the desired state.

In some implementations, the training signal r further may comprisedesired output spike train yd, current performance may be effected, atleast partly, by transition from prior state to the current state; andthe reinforcement learning may be configured based at least in part onthe reinforcement spiking indicator so that: positive reinforcement whenthe current performance may be closer to the desired performance, andthe reinforcement may be negative when the current performance may befarther from the desired performance.

In some implementations, the stochastic learning block may be operableaccording to stochastic process characterized by current state anddesired state, the process being described by at least state variableconfigured to transition the learning block from current state to thedesired state; the hybrid learning rule may be characterized by a hybridperformance function F comprising a simultaneous combination ofreinforcement learning performance function Fre and supervised learningperformance function Fsu; and the simultaneous combination may beeffectuated by at least in part on a value of the hybrid performancefunction F determined at a time step t, the value comprisingreinforcement performance function Fre value and supervised learningperformance function Fsu value.

These and other objects, features, and characteristics of the presentdisclosure, as well as the methods of operation and functions of therelated elements of structure and the combination of parts and economiesof manufacture, will become more apparent upon consideration of thefollowing description and the appended claims with reference to theaccompanying drawings, all of which form a part of this specification,wherein like reference numerals designate corresponding parts in thevarious figures. It is to be expressly understood, however, that thedrawings are for the purpose of illustration and description only andare not intended as a definition of the limits of the disclosure. Asused in the specification and in the claims, the singular form of “a”,“an”, and “the” include plural referents unless the context clearlydictates otherwise.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating a typical architecture of anadaptive system according to prior art.

FIG. 1A is a block diagram illustrating multi-task learning controllerapparatus according to prior art.

FIG. 2 is a graphical illustration of typical input and output spiketrains according to prior art.

FIG. 3 is a block diagram illustrating generalized learning apparatus,in accordance with one or more implementations.

FIG. 4 is a block diagram illustrating learning block apparatus of FIG.3, in accordance with one or more implementations.

FIG. 4A is a block diagram illustrating exemplary implementations ofperformance determination block of the learning block apparatus of FIG.4, in accordance with the disclosure.

FIG. 5 is a block diagram illustrating generalized learning apparatus,in accordance with one or more implementations.

FIG. 5A is a block diagram illustrating generalized learning blockconfigured for implementing different learning rules, in accordance withone or more implementations.

FIG. 58 is a block diagram illustrating generalized learning blockconfigured for implementing different learning rules, in accordance withone or more implementations.

FIG. 6A is a block diagram illustrating a spiking neural network,comprising three dynamically configured partitions, configured toeffectuate generalized learning block of FIG. 4, in accordance with oneor more implementations.

FIG. 6B is a block diagram illustrating a spiking neural network,comprising two dynamically configured partitions, adapted to effectuategeneralized learning, in accordance with one or more implementations.

FIG. 7 is a block diagram illustrating spiking neural network configuredto effectuate multiple learning rules, in accordance with one or moreimplementations.

FIG. 8A is a logical flow diagram illustrating generalized learningmethod for use with the apparatus of FIG. 5A, in accordance with one ormore implementations.

FIG. 8B is a logical flow diagram illustrating dynamic reconfigurationmethod for use with the apparatus of FIG. 5A, in accordance with one ormore implementations.

FIG. 9A is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7 prior to learning, in accordance with oneor more implementations, where data in the panels from top to bottomcomprise: (i) input spike pattern; (ii) output activity of the networkbefore learning; (iii) supervisor spike pattern; (iv) positivereinforcement spike pattern; and (v) negative reinforcement spikepattern.

FIG. 9B is a plot presenting simulations data illustrating supervisedlearning operation of the neural network of FIG. 7, in accordance withone or more implementations, where data in the panels from top to bottomcomprise: (i) input spike pattern; (ii) output activity of the networkbefore learning; (iii) supervisor spike pattern; (iv) positivereinforcement spike pattern; and (v) negative reinforcement spikepattern.

FIG. 9C is a plot presenting simulations data illustrating reinforcementlearning operation of the neural network of FIG. 7, in accordance withone or more implementations, where data in the panels from top to bottomcomprise: (i) input spike pattern; (ii) output activity of the networkafter learning; (iii) supervisor spike pattern; (iv) positivereinforcement spike pattern; and (v) negative reinforcement spikepattern.

FIG. 9D is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising reinforcement learning aidedwith small portion of supervisor spikes, in accordance with one or moreimplementations, where data in the panels from top to bottom comprise:(i) input spike pattern; (ii) output activity of the network afterlearning; (iii) supervisor spike pattern; (iv) positive reinforcementspike pattern; and (v) negative reinforcement spike pattern.

FIG. 9E is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising an equal mix of reinforcementand supervised learning signals, in accordance with one or moreimplementations, where data in the panels from top to bottom comprise:(i) input spike pattern; (ii) output activity of the network afterlearning; (iii) supervisor spike pattern; (iv) positive reinforcementspike pattern; and (v) negative reinforcement spike pattern.

FIG. 9F is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising supervised learning augmentedwith a 50% fraction of reinforcement spikes, in accordance with one ormore implementations, where data in the panels from top to bottomcomprise: (i) input spike pattern; (ii) output activity of the networkafter learning; (iii) supervisor spike pattern; (iv) positivereinforcement spike pattern; and (v) negative reinforcement spikepattern.

FIG. 10A is a plot presenting simulations data illustrating supervisedlearning operation of the neural network of FIG. 7, in accordance withone or more implementations, where data in the panels from top to bottomcomprise: (i) input spike pattern; (ii) output activity of the networkbefore learning; (iii) supervisor spike pattern.

FIG. 10B is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising supervised learning augmentedby a small amount of unsupervised learning, modeled as 15% fraction ofrandomly distributed (Poisson) spikes, in accordance with one or moreimplementations, where data in the panels from top to bottom comprise:(i) input spike pattern; (ii) output activity of the network afterlearning, (iii) supervisor spike pattern.

FIG. 10C is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising supervised learning augmentedby a substantial amount of unsupervised learning, modeled as 80%fraction of Poisson spikes, in accordance with one or moreimplementations, where data in the panels from top to bottom comprise:(i) input spike pattern; (ii) output activity of the network afterlearning, (iii) supervisor spike pattern.

FIG. 11 is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising supervised learning andreinforcement learning, augmented by a small amount of unsupervisedlearning, modeled as 15% fraction of Poisson spikes, in accordance withone or more implementations, where data in the panels from top to bottomcomprise: (i) input spike pattern; (ii) output activity of the networkafter learning, (iii) supervisor spike pattern; (iv) positivereinforcement spike pattern; and (v) negative reinforcement spikepattern.

FIG. 12 is a plot presenting simulations data illustrating supervisedlearning operation of the spiking neural network of FIG. 7. Data in thepanels from top to bottom comprise: (i) input spike pattern; (ii) outputactivity of the network after learning; (iii) supervisor spike pattern.

FIG. 13 is a plot presenting simulations data illustrating predictivesupervised learning operation of the spiking neural network of FIG. 7.Data in the panels from top to bottom comprise: (i) input spike pattern;(ii) output activity of the network after learning; (iii) supervisorspike pattern.

FIG. 14 is a plot presenting simulations data illustrating reciprocalsupervised learning operation of the spiking neural network of FIG. 7.Data in the panels from top to bottom comprise: (i) input spike pattern;(ii) output activity of the network after learning; (iii) supervisorspike pattern.

FIG. 15A is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising unsupervised learning. Data inthe panels from top to bottom comprise: (i) input spike pattern; (ii)output activity of the network after learning; (iii) evolution ofweights during learning.

FIG. 15B is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising unsupervised learning viaKullback-Liebler divergence minimization. Data in the top panelsrepresents the average performance, while data in the bottom panel showsevolution of weights during learning.

FIG. 15C is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising unsupervised learning viaKullback-Liebler divergence minimization. Data in the top panelsrepresents the input spike pattern; while data in the bottom panel showsnetwork output after learning.

FIG. 16 is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising reinforcement learning. Data inthe top panel illustrate mean distance <d> between the actual positionof an AUV y(t) and the desired position of the AUV y^(d)(t); while datain the bottom panel present variance of the distance d.

FIG. 17A is a plot presenting simulations data illustrating operation ofthe neural network of FIG. 7, comprising reinforcement learning. Data inthe panels from top to bottom comprise: (i) input spike pattern; (ii)output activity of the network at time interval corresponding to theepoch 250 of FIG. 17B, (iii) reward spike pattern.

FIG. 17B is a plot presenting simulations data illustrating averagedperformance (top) and evolution of weights (bottom) of the spikingneural network of FIG. 7, comprising reinforcement learning configuredin accordance with one or more implementations.

FIG. 18 is a logical flow diagram illustrating automatic computation ofeligibility traces in a spiking neural network, in accordance with oneor more implementations.

FIG. 19 is a program listing illustrating textual description of spikingneuron dynamics and stochastic properties configured for processing byMatlab® symbolic computation engine in order to automatically generatescore function, in accordance with one or more implementations.

All Figures disclosed herein are © Copyright 2012 Brain Corporation. Allrights reserved.

DETAILED DESCRIPTION

Exemplary implementations of the present disclosure will now bedescribed in detail with reference to the drawings, which are providedas illustrative examples so as to enable those skilled in the art topractice the disclosure. Notably, the figures and examples below are notmeant to limit the scope of the present disclosure to a singleimplementation, but other implementations are possible by way ofinterchange of or combination with some or all of the described orillustrated elements. Wherever convenient, the same reference numberswill be used throughout the drawings to refer to same or similar parts.

Where certain elements of these implementations can be partially orfully implemented using known components, only those portions of suchknown components that are necessary for an understanding of the presentdisclosure will be described, and detailed descriptions of otherportions of such known components will be omitted so as not to obscurethe disclosure.

In the present specification, an implementation showing a singularcomponent should not be considered limiting; rather, the disclosure isintended to encompass other implementations including a plurality of thesame component, and vice-versa, unless explicitly stated otherwiseherein.

Further, the present disclosure encompasses present and future knownequivalents to the components referred to herein by way of illustration.

As used herein, the term “bus” is meant generally to denote all types ofinterconnection or communication architecture that is used to access thesynaptic and neuron memory. The “bus” may be optical, wireless,infrared, and/or another type of communication medium. The exacttopology of the bus could be for example standard “bus”, hierarchicalbus, network-on-chip, address-event-representation (AER) connection,and/or other type of communication topology used for accessing, e.g.,different memories in pulse-based system.

As used herein, the terms “computer”, “computing device”, and“computerized device “may include one or more of personal computers(PCs) and/or minicomputers (e.g., desktop, laptop, and/or other PCs),mainframe computers, workstations, servers, personal digital assistants(PDAs), handheld computers, embedded computers, programmable logicdevices, personal communicators, tablet computers, portable navigationaids, J2ME equipped devices, cellular telephones, smart phones, personalintegrated communication and/or entertainment devices, and/or any otherdevice capable of executing a set of instructions and processing anincoming data signal.

As used herein, the term “computer program” or “software” may includeany sequence of human and/or machine cognizable steps which perform afunction. Such program may be rendered in a programming language and/orenvironment including one or more of C/C++, C#, Fortran, COBOL, MATLAB™,PASCAL, Python, assembly language, markup languages (e.g., HTML, SGML,XML, VoXML), object-oriented environments (e.g., Common Object RequestBroker Architecture (CORBA)), Java™ (e.g., J2ME, Java Beans), BinaryRuntime Environment (e.g., BREW), and/or other programming languagesand/or environments.

As used herein, the terms “connection”, “link”, “transmission channel”,“delay line”, “wireless” may include a causal link between any two ormore entities (whether physical or logical/virtual), which may enableinformation exchange between the entities.

As used herein, the term “memory” may include an integrated circuitand/or other storage device adapted for storing digital data. By way ofnon-limiting example, memory may include one or more of ROM, PROM,EEPROM, DRAM, Mobile DRAM, SDRAM, DDR/2 SDRAM, EDO/FPMS, RLDRAM, SRAM,“flash” memory (e.g., NAND/NOR), memristor memory, PSRAM, and/or othertypes of memory.

As used herein, the terms “integrated circuit”, “chip”, and “IC” aremeant to refer to an electronic circuit manufactured by the patterneddiffusion of trace elements into the surface of a thin substrate ofsemiconductor material. By way of non-limiting example, integratedcircuits may include field programmable gate arrays (e.g., FPGAs), aprogrammable logic device (PLD), reconfigurable computer fabrics (RCFs),application-specific integrated circuits (ASICs), and/or other types ofintegrated circuits.

As used herein, the terms “microprocessor” and “digital processor” aremeant generally to include digital processing devices. By way ofnon-limiting example, digital processing devices may include one or moreof digital signal processors (DSPs), reduced instruction set computers(RISC), general-purpose (CISC) processors, microprocessors, gate arrays(e.g., field programmable gate arrays (FPGAs)), PLDs, reconfigurablecomputer fabrics (RCFs), array processors, secure microprocessors,application-specific integrated circuits (ASICs), and/or other digitalprocessing devices. Such digital processors may be contained on a singleunitary IC die, or distributed across multiple components.

As used herein, the term “network interface” refers to any signal, data,and/or software interface with a component, network, and/or process. Byway of non-limiting example, a network interface may include one or moreof FireWire (e.g., FW400, FW800, etc.), USB (e.g., USB2), Ethernet(e.g., 10/100, 10/100/1000 (Gigabit Ethernet), 10-Gig-E, etc.), MoCA,Coaxsys (e.g., TVnet™), radio frequency tuner (e.g., in-band or OOB,cable modem, etc.), Wi-Fi (802.11), WiMAX (802.16), PAN (e.g., 802.15),cellular (e.g., 3G, LTE/LTE-A/TD-LTE, GSM, etc.), IrDA families, and/orother network interfaces.

As used herein, the terms “node”, “neuron”, and “neuronal node” aremeant to refer, without limitation, to a network unit (e.g., a spikingneuron and a set of synapses configured to provide input signals to theneuron) having parameters that are subject to adaptation in accordancewith a model.

As used herein, the terms “state” and “node state” is meant generally todenote a full (or partial) set of dynamic variables used to describenode state.

As used herein, the term “synaptic channel”, “connection”, “link”,“transmission channel”, “delay line”, and “communications channel”include a link between any two or more entities (whether physical (wiredor wireless), or logical/virtual) which enables information exchangebetween the entities, and may be characterized by a one or morevariables affecting the information exchange.

As used herein, the term “Wi-Fi” includes one or more of IEEE-Std.802.11, variants of IEEE-Std. 802.11, standards related to IEEE-Std.802.11 (e.g., 802.11a/b/g/n/s/v), and/or other wireless standards.

As used herein, the term “wireless” means any wireless signal, data,communication, and/or other wireless interface. By way of non-limitingexample, a wireless interface may include one or more of Wi-Fi,Bluetooth, 3G (3GPP/3GPP2), HSDPA/HSUPA, TDMA, CDMA (e.g., IS-95A,WCDMA, etc.), FHSS, DSSS, GSM, PAN/802.15, WiMAX (802.16), 802.20,narrowband/FDMA, OFDM, PCS/DCS, LTE/LTE-A/TD-LTE, analog cellular, CDPD,satellite systems, millimeter wave or microwave systems, acoustic,infrared (i.e., IrDA), and/or other wireless interfaces.

Overview

The present disclosure provides, among other things, a computerizedapparatus and methods for implementing generalized learning rules givenmultiple cost measures. In one implementation of the disclosure,adaptive spiking neuron network signal processing system may flexiblycombine different learning rules (e.g., supervised, unsupervised,reinforcement learning, and/or other learning rules) with differentmethods (e.g., online, batch, and/or other learning methods). Thegeneralized learning apparatus of the disclosure may employ, in someimplementations, modular architecture where learning tasks are separatedfrom control tasks, so that changes in one of the blocks do notnecessitate changes within the other block. By separating implementationof learning tasks from the control tasks, the framework may furtherallow dynamic reconfiguration of the learning block in response to atask change or learning method change in real time. The generalizedlearning apparatus may be capable of implementing several learning rulesconcurrently based on the desired control task and without requiringusers to explicitly identify the required learning rule composition forthat application.

The generalized learning framework described herein advantageouslyprovides for learning implementations that do not affect regularoperation of the signal system (e.g., processing of data). Hence, a needfor a separate learning stage may be obviated so that learning may beturned off and on again when appropriate.

One or more generalized learning methodologies described herein mayenable different parts of the same network to implement differentadaptive tasks. The end user of the adaptive device may be enabled topartition network into different parts, connect these partsappropriately, and assign cost functions to each task (e.g., selectingthem from predefined set of rules or implementing a custom rule). A usermay not be required to understand detailed implementation of theadaptive system (e.g., plasticity rules, neuronal dynamics, etc.) nor ishe required to be able to derive the performance function and determineits gradient for each learning task. Instead, a user may be able tooperate generalized learning apparatus of the disclosure by assigningtask functions and connectivity map to each partition.

Generalized Learning Apparatus

Detailed descriptions of various implementations of apparatuses andmethods of the disclosure are now provided. Although certain aspects ofthe disclosure may be understood in the context of robotic adaptivecontrol system comprising a spiking neural network, the disclosure isnot so limited. Implementations of the disclosure may also be used forimplementing a variety of learning systems, such as, for example, signalprediction (e.g., supervised learning), finance applications, dataclustering (e.g., unsupervised learning), inventory control, datamining, and/or other applications that do not require performancefunction derivative computations.

Implementations of the disclosure may be, for example, deployed in ahardware and/or software implementation of a neuromorphic computersystem. In some implementations, a robotic system may include aprocessor embodied in an application specific integrated circuit, whichcan be adapted or configured for use in an embedded application (e.g., aprosthetic device).

FIG. 3 illustrates one exemplary learning apparatus useful to thedisclosure. The apparatus 300 shown in FIG. 3 comprises the controlblock 310, which may include a spiking neural network configured tocontrol a robotic arm and may be parameterized by the weights ofconnections between artificial neurons, and learning block 320, whichmay implement learning and/or calculating the changes in the connectionweights. The control block 310 may receive an input signal x, and maygenerate an output signal y. The output signal y may include motorcontrol commands configured to move a robotic arm along a desiredtrajectory. The control block 310 may be characterized by a system modelcomprising system internal state variables S. An internal state variableS may include a membrane voltage of the neuron, conductance of themembrane, and/or other variables. The control block 310 may becharacterized by learning parameters w, which may include synapticweights of the connections, firing threshold, resting potential of theneuron, and/or other parameters. In one or more implementations, theparameters w may comprise probabilities of signal transmission betweenthe units (e.g., neurons) of the network.

The input signal x(t) may comprise data used for solving a particularcontrol task. In one or more implementations, such as those involving arobotic arm or autonomous robot, the signal x(t) may comprise a streamof raw sensor data (e.g., proximity, inertial, terrain imaging, and/orother raw sensor data) and/or preprocessed data (e.g., velocity,extracted from accelerometers, distance to obstacle, positions, and/orother preprocessed data). In some implementations, such as thoseinvolving object recognition, the signal x(t) may comprise an array ofpixel values (e.g., RGB, CMYK, HSV, HSL, grayscale, and/or other pixelvalues) in the input image, or preprocessed data (e.g., levels ofactivations of Gabor filters for face recognition, contours, and/orother preprocessed data). In one or more implementations, the inputsignal x(t) may comprise desired motion trajectory, for example, inorder to predict future state of the robot on the basis of current stateand desired motion.

The control block 310 of FIG. 3 may comprise a probabilistic dynamicsystem, which may be characterized by an analytical input-output (x→y)probabilistic relationship having a conditional probability distributionassociated therewith:

P=p(y|x,w)  (Eqn. 11)

In Eqn. 11, the parameter w may denote various system parametersincluding connection efficacy, firing threshold, resting potential ofthe neuron, and/or other parameters. The analytical relationship of Eqn.1 may be selected such that the gradient of ln [p(y|x,w)] with respectto the system parameter w exists and can be calculated. The frameworkshown in FIG. 3 may be configured to estimate rules for changing thesystem parameters (e.g., learning rules) so that the performancefunction F(x,y,r) is minimized for the current set of inputs and outputsand system dynamics S.

In some implementations, the control performance function may beconfigured to reflect the properties of inputs and outputs (x,y). Thevalues F(x,y,r) may be calculated directly by the learning block 320without relying on external signal r when providing solution ofunsupervised learning tasks.

In some implementations, the value of the function F may be calculatedbased on a difference between the output y of the control block 310 anda reference signal y^(d) characterizing the desired control blockoutput. This configuration may provide solutions for supervised learningtasks, as described in detail below.

In some implementations, the value of the performance function F may bedetermined based on the external signal r. This configuration mayprovide solutions for reinforcement learning tasks, where r representsreward and punishment signals from the environment.

Learning Block

The learning block 320 may implement learning framework according to theimplementation of FIG. 3 that enables generalized learning methodswithout relying on calculations of the performance function F derivativein order to solve unsupervised, supervised, reinforcement, and/or otherlearning tasks. The block 320 may receive the input x and output ysignals (denoted by the arrow 302_1, 308_1, respectively, in FIG. 3), aswell as the state information 305. In some implementations, such asthose involving supervised and reinforcement learning, external teachingsignal r may be provided to the block 320 as indicated by the arrow 304in FIG. 3. The teaching signal may comprise, in some implementations,the desired motion trajectory, and/or reward and punishment signals fromthe external environment.

In one or more implementations the learning block 320 may optimizeperformance of the control system (e.g., the system 300 of FIG. 3) thatis characterized by minimization of the average value of the performancefunction F(x,y,r) as described in detail in co-owned and co-pending U.S.patent application Ser. No. 13/XXX,XXX entitled “STOCHASTIC APPARATUSAND METHODS FOR IMPLEMENTING GENERALIZED LEARNING RULES”, incorporatedsupra. The above-referenced application describes, in one or moreimplementations, minimizing the average performance

F

_(x,y,r) using, for example, gradient descend algorithms where

$\begin{matrix}{{\frac{\partial}{\partial w_{i}}{\langle{F\left( {x,y,r} \right)}\rangle}_{x,y,r}} = {\langle{{\langle{{F\left( {x,y,r} \right)}\frac{\partial\;}{\partial w_{i}}{\ln\left( {p\left( {y\left. {x,w} \right)} \right)}\rangle \right.}_{x,y}}\rangle}_{r}{where}\text{:}}}} & \left( {{Eqn}.\mspace{11mu} 12} \right) \\{- {\ln\left( {{p\left( {y\left. {x,w} \right)} \right)} = {h\left( {y\left. {x,w} \right)} \right.}} \right.}} & \left( {{Eqn}.\mspace{11mu} 13} \right)\end{matrix}$

is the per-stimulus entropy of the system response (or ‘surprisal’). Theprobability of the external signal p(r|x,y) may be characteristic of theexternal environment and may not change due to adaptation. That propertymay allow omission of averaging over external signals r in subsequentconsideration of learning rules.

As illustrated in FIG. 3, the learning block may have access to thesystem's inputs and outputs, and/or system internal state S. In someimplementations, the learning block may be provided with additionalinputs 304 (e.g., reinforcement signals, desired output, and/or currentcosts of control movements, etc.) that are related to the current taskof the control block.

The learning block may estimate changes of the system parameters w thatminimize the performance function F, and may provide the parameteradjustment information Δw to the control block 310, as indicated by thearrow 306 in FIG. 3. In some implementations, the learning block may beconfigured to modify the learning parameters w of the controller block.In one or more implementations (not shown), the learning block may beconfigured to communicate parameters w (as depicted by the arrow 306 inFIG. 3) for further use by the controller block 310, or to anotherentity (not shown).

By separating learning related tasks into a separate block (e.g., theblock 320 in FIG. 3) from control tasks, the architecture shown in FIG.3 may provide flexibility of applying different (or modifying) learningalgorithms without requiring modifications in the control block model.In other words, the methodology illustrated in FIG. 3 may enableimplementation of the learning process in such a way that regularfunctionality of the control aspects of the system 300 is not affected.For example, learning may be turned off and on again as required withthe control block functionality being unaffected.

The detailed structure of the learning block 420 is shown and describedwith respect to FIG. 4. The learning block 420 may comprise one or moreof gradient determination (GD) block 422, performance determination (PD)block 424 and parameter adaptation block (PA) 426, and/or othercomponents. The implementation shown in FIG. 4 may decompose thelearning process of the block 420 into two parts. Atask-dependent/system independent part (i.e., the block 420) mayimplement a performance determination aspect of learning that isdependent only on the specified learning task (e.g., supervised).Implementation of the PD block 424 may not depend on particulars of thecontrol block (e.g., block 310 in FIG. 3) such as, for example, neuralnetwork composition, neuron operating dynamics, and/or otherparticulars). The second part of the learning block 420, comprised ofthe blocks 422 and 426 in FIG. 4, may implement task-independent/systemdependent aspects of the learning block operation. The implementation ofthe GD block 422 and PA block 426 may be the same for individuallearning rules (e.g., supervised and/or unsupervised). The GD blockimplementation may further comprises particulars of gradientdetermination and parameter adaptation that are specific to thecontroller system 310 architecture (e.g., neural network composition,neuron operating dynamics, and/or plasticity rules). The architectureshown in FIG. 4 may allow users to modify task-specific and/orsystem-specific portions independently from one another, therebyenabling flexible control of the system performance. An advantage of theframework may be that the learning can be implemented in a way that doesnot affect the normal protocol of the functioning of the system (exceptof changing the parameters w). For example, there may be no need in aseparate learning stage and learning may be turned off and on again whenappropriate.

Gradient Determination Block

The GD block may be configured to determine the score function g by,inter alia, computing derivatives of the logarithm of the conditionalprobability with respect to the parameters that are subjected to changeduring learning based on the current inputs x, outputs y, and statevariables S, denoted by the arrows 402, 408, 410, respectively, in FIG.4. The GD block may produce an estimate of the score function g, denotedby the arrow 418 in FIG. 4 that is independent of the particularlearning task, (e.g., reinforcement, unsupervised, and/or supervisedearning). In some implementations, where the learning model comprisesmultiple parameters w_(i), the score function g may be represented as avector g, comprising scores g_(i) associated with individual parametercomponents w_(i).

Implementation of this block may be non-trivial for the complex adaptivesystems, such as spiking neural networks. However, using the frameworkdescribed herein, this implementation may need to be changed only onceand then used without changing for different learning tasks, asdescribed in detail below.

In order to apply SF/LR methods for spiking neurons, a score function

$g_{i} \equiv \frac{\partial{h\left( {y\left. x \right)} \right.}}{\partial w_{i}}$

may be calculated for individual spiking neurons parameters to bechanged. If spiking patterns are viewed on finite interval length T asan input x and output y of the neuron, then the score function may takethe following form:

$\begin{matrix}{g_{i} = {\frac{\partial{h\left( {y_{T}\left. x_{T} \right)} \right.}}{\partial w_{i}} = {{- {\sum\limits_{t_{l} \in y_{T}}{\frac{1}{\lambda \left( t_{l} \right)}\frac{\partial{\lambda \left( t_{l} \right)}}{\partial w_{i}}}}} + {\int_{T}{\frac{\partial{\lambda (s)}}{\partial w_{i}}{{s}.}}}}}} & \left( {{Eqn}.\mspace{11mu} 14} \right)\end{matrix}$

where time moments t_(i) belong to neuron's output pattern y_(T) (neurongenerates spike at these time moments).

If an output of the neuron at each time moment is considered (e.g.,whether there is an output spike or not), then an instantaneous value ofthe score function may be calculated that is a time derivative of theinterval score function:

$\begin{matrix}{g_{i} = {\frac{\partial{h\left( {{y(t)}\left. x \right)} \right.}}{\partial w_{i}} = {\frac{\partial{\lambda (t)}}{\partial w_{i}}\left( {1 - {\sum\limits_{t_{l}}\frac{\delta \left( {t - t_{l}} \right)}{\lambda (t)}}} \right)}}} & \left( {{Eqn}.\mspace{11mu} 15} \right)\end{matrix}$

where t_(l) is times of output spikes and δ(t) is a delta function.

For discrete time the score function for spiking pattern on interval Tmay be calculated as:

$\begin{matrix}{g_{i} = {\frac{\partial{h\left( {y_{T}\left. x_{T} \right)} \right.}}{\partial w_{i}} = {{- {\sum\limits_{t_{i} \in y_{T}}{\frac{1 - {\Lambda \left( t_{i} \right)}}{\Lambda \left( t_{i} \right)}\frac{\partial{\lambda \left( t_{i} \right)}}{\partial w_{i}}\Delta \; t}}} + {\sum\limits_{t_{i} \notin y_{T}}{{\frac{\partial{\lambda \left( t_{i} \right)}}{\partial w_{i}} \cdot \Delta}\; t}}}}} & \left( {{Eqn}.\mspace{11mu} 16} \right)\end{matrix}$

where t_(l)εy_(T) denotes time steps when neuron generated a spike.

Instantaneous value of the score function in discrete time may equal:

$\begin{matrix}{g_{i} = {\frac{\partial h_{\Delta \; t}}{\partial w_{i}} = {\frac{\partial\lambda}{\partial w_{i}}\left( {1 - {\sum_{l}{\frac{1 - {\Lambda (t)}}{\Lambda (t)}{\delta_{d}\left( {t - t_{l}} \right)}}}} \right)}}} & \left( {{Eqn}.\mspace{11mu} 17} \right)\end{matrix}$

where t_(l) is the times of output spikes andδ_(d l (t) is the Kronecker delta.)

In order to calculate the score function,

$\frac{\partial{\lambda (t)}}{\partial w_{i}}$

may be calculated, which is a derivative of the instantaneousprobability density with respect to a learning parameter w_(i) of thei-th neuron. Without loss of generality, two cases of learning areconsidered below: input weights learning (synaptic plasticity) andstochastic threshold tuning (intrinsic plasticity). A derivative ofother less common parameters of the neuron model (e.g., membrane,synaptic dynamic, and/or other constants) may be calculated.

The neuron may receive n input spiking channels. External current to theneuron I^(ext) in the neuron's dynamic equation may be modeled Eqn. 6 asa sum of filtered and weighted input spikes from all input channels:

$\begin{matrix}{I^{ext} = {\sum\limits_{i}^{n}{\sum\limits_{t_{j}^{i} \in x^{i}}{w_{i}{ɛ\left( {t - t_{j}^{i}} \right)}}}}} & \left( {{Eqn}.\mspace{11mu} 18} \right)\end{matrix}$

where: i is the index of the input channel; x^(i) is the stream of inputspikes on the i-th channel; t_(j) ^(i) is the times of input spikes inthe i-th channel; w_(i) is the weight of the i-th channel; and ε(t) is ageneric function that models post-synaptic currents from input spikes.In some implementations, the post-synaptic current function may beconfigured as: ε(t)≡δ(t), ε(t)≡e^(−t/t) ^(s) H(t), where δ(t) is a deltafunction, H(t) is a Heaviside function, and τ_(s) is a synaptic timeconstant.

A derivative of instantaneous probability density with respect to thei-th channel's weight may be taken using chain rule:

$\begin{matrix}{\frac{\partial\lambda}{\partial w_{i}} = {\sum\limits_{j}{\left( {\frac{\partial\lambda_{i}}{\partial q_{j}} \cdot {\nabla_{w_{i}}q_{j}}} \right)\mspace{14mu} {where}\mspace{14mu} \frac{\partial\lambda}{\partial\overset{r}{q}}}}} & \left( {{Eqn}.\mspace{14mu} 19} \right)\end{matrix}$

is a vector of derivatives of instantaneous probability density withrespect to the state variable; and

S _(i)(t)=∇_(w) _(i) {right arrow over (q)}  (Eqn. 20)

is the gradient of the neuron internal state with respect to the i^(th)weight (also referred to as the i-th state eligibility trace). In orderto determine the state eligibility trace of Eqn. 20 for generalizedneuronal model, such as, for example, described by equations Eqn. 6 andEqn. 18, derivative with respect to the learning weight w_(i) may bedetermined as:

$\begin{matrix}{{\frac{\partial}{\partial w_{i}}\left( \frac{\overset{\rightarrow}{q}}{t} \right)} = {{\frac{\partial}{\partial w_{i}}\left( {V\left( \overset{\rightarrow}{q} \right)} \right)} + {\frac{\partial\;}{\partial w_{i}}\left( {\sum_{t^{out}}{{R\left( \overset{\rightarrow}{q} \right)}{\delta \left( {t - t^{out}} \right)}}} \right)} + {\frac{\partial\;}{\partial w_{i}}\left( {{G\left( \overset{\rightarrow}{q} \right)}I^{ext}} \right)}}} & \left( {{Eqn}.\mspace{11mu} 21} \right)\end{matrix}$

The order in which the derivatives in the left side of the equations aretaken may be changed, and then the chain rule may be used to obtain thefollowing equations (arguments of evolution functions are omitted):

$\begin{matrix}{{\frac{{S_{i}(t)}}{t} = {{\left( {{{Jv}\left( \overset{\rightarrow}{q} \right)} + {{J_{G}\left( \overset{\rightarrow}{q} \right)} \cdot I^{ext}}} \right) \cdot S_{i}} + {\sum\limits_{t^{out}}{{J_{R}\left( \overset{\rightarrow}{q} \right)} \cdot S_{i} \cdot {\delta \left( {t - t^{out}} \right)}}} + {{G\left( \overset{\rightarrow}{q} \right)}{\sum_{t_{j}^{i} \in x^{j}}{ɛ\left( {t - t_{j}^{i}} \right)}}}}},} & \left( {{Eqn}.\mspace{11mu} 22} \right)\end{matrix}$

Where J_(F), J_(R), J_(G) are Jacobian matrices of the respectiveevolution functions V, R, G.

As an example, evaluating Jacobean matrices IF neuron may produce:

J _(V) =−C;J _(R)=−1;G({right arrow over (q)})=1;J _(G)=0,  (Eqn. 23)

so Eqn. 22 for the i-th state eligibility trace may take the followingform:

$\begin{matrix}{{\frac{}{t}u_{w_{i}}} = {{- {Cu}_{w_{i}}} - {\sum\limits_{t^{out}}{u_{w_{i}} \cdot {\delta \left( {t - t^{out}} \right)}}} + {\sum\limits_{t_{j}^{i} \in x^{i}}{ɛ\left( {t - t_{j}^{i}} \right)}}}} & \left( {{Eqn}.\mspace{14mu} 24} \right)\end{matrix}$

where u_(w) _(j) denotes derivative of the state variable (e.g.,voltage) with respect to the i-th weight.

A solution of Eqn. 24 may represent post-synaptic potential for the i-thunit and may be determined as a sum of all received input spikes at theunit (e.g., a neuron), where the unit is reset to zero after each outputspike:

$\begin{matrix}{u_{w_{i}} = {{\sum\limits_{t_{j}^{i} \in x^{i}}{\int_{- \infty}^{t}{^{{- {({t - \tau})}}C}{ɛ\left( {\tau - t_{j}^{i}} \right)}}}} = {\sum\limits_{t_{j}^{i} \in x^{i}}{\alpha \left( {t - t_{j}^{i}} \right)}}}} & \left( {{Eqn}.\mspace{14mu} 25} \right)\end{matrix}$

where α(t) is the post-synaptic potential (PSP) from the j^(th) inputspike.

Applying the framework of Eqn. 22-Eqn. 25 to a previously describedneuronal (hereinafter IZ neuronal), the Jacobian matrices of therespective evolution functions F, R, G may be expressed as:

$\begin{matrix}{{J_{V} = \begin{pmatrix}{{0.08{v(t)}} + 5} & {- 1} \\{ab} & a\end{pmatrix}};\mspace{14mu} {J_{R} = \begin{pmatrix}{- 1} & 0 \\0 & 0\end{pmatrix}};\mspace{14mu} {{G\left( \overset{->}{q} \right)} = \begin{pmatrix}1 \\0\end{pmatrix}};\mspace{14mu} {J_{G} = \begin{pmatrix}0 \\0\end{pmatrix}}} & \left( {{Eqn}.\mspace{14mu} 26} \right)\end{matrix}$

The IZ neuronal model may further be characterized using two first-ordernonlinear differential equations describing time evolution of synapticweights associated with each input interface (e.g., pre-synapticconnection) of a neuron, in the following form:

$\begin{matrix}{{{\frac{}{t}v_{w_{i}}} = {{\left( {{0.08v} + 5} \right)v_{w_{i}}} - u_{w_{i}} - {\sum\limits_{t^{out}}{u_{w_{i}} \cdot {\delta \left( {t - t^{out}} \right)}}} + {\sum\limits_{t_{j}^{i} \in x^{i}}{ɛ\left( {t - t_{j}^{i}} \right)}}}}\mspace{79mu} {{\frac{}{t}u_{w_{i}}} = {{abv}_{w_{i}} - {a\; u_{w_{i}}}}}} & \left( {{Eqn}.\mspace{14mu} 27} \right)\end{matrix}$

When using the exponential stochastic threshold configured as:

λ=λ₀ e ^(κ(v(t)-θ)),  (Eqn. 28)

then the derivative of the IPD for IZ neuronal neuron becomes:

$\begin{matrix}{\frac{\partial\lambda}{\partial w_{i}} = {v_{w_{i}}\kappa \; {{\lambda (t)}.}}} & \left( {{Eqn}.\mspace{14mu} 29} \right)\end{matrix}$

If we use the exponential stochastic threshold Eqn. 2, the finalexpression for the derivative of instantaneous probability

$\frac{\partial{\lambda (t)}}{\partial w}$

for IF neuron becomes:

$\begin{matrix}{\frac{\partial\lambda}{\partial w_{i}} = {{\frac{\partial\lambda}{\partial u}\frac{\partial u}{\partial w_{i}}} = {\kappa \; {\lambda (t)}{\sum\limits_{t_{j}^{i} \in x^{i}}{\alpha \left( {t - t_{j}^{i}} \right)}}}}} & \left( {{Eqn}.\mspace{14mu} 30} \right)\end{matrix}$

Combining Eqn. 30 with Eqn. 15 and Eqn. 17 we obtain score functionvalues for the stochastic Integrate-and-Fire neuron in continuoustime-space as:

$\begin{matrix}{g_{i} = {\frac{\partial{h\left( {y(t)} \middle| x \right)}}{\partial w_{i}} = {\kappa {\sum\limits_{t_{j}^{i} \in x^{i}}{{\alpha \left( {t - t_{j}^{i}} \right)}\left( {{\lambda (t)} - {\sum\limits_{t^{out} \in y}{\delta \left( {t - t^{out}} \right)}}} \right)}}}}} & \left( {{Eqn}.\mspace{14mu} 31} \right)\end{matrix}$

and in discrete time:

$\begin{matrix}{{g_{i} = {\frac{\partial{h_{\Delta \; t}\left( {y(t)} \middle| x \right)}}{\partial w_{i}} = {{{\kappa\lambda}(t)}{\sum\limits_{t_{j}^{i} \in x^{i}}{{\alpha \left( {t - t_{j}^{i}} \right)}\left( {1 - {\sum\limits_{t^{out} \in y}\frac{\delta_{d}\left( {t - t^{out}} \right)}{\Lambda (t)}}} \right)\Delta \; t}}}}}\;} & \left( {{Eqn}.\mspace{14mu} 32} \right)\end{matrix}$

In one or more implementations, the gradient determination block may beconfigured to determine the score function g based on particular inputsinto the neuron(s), neuron outputs, and internal neuron state,according, for example with Eqn. 15. Furthermore, in someimplementations, using the methodology described herein and providingdescription of neurons dynamics and stochastic properties in textualform, as shown and described in detail with respect to FIG. 19 below,advantageously allows the use of analytical mathematics computer aideddesign (CAD) tools in order to automatically obtain score function, suchas for example Eqn. 32.

Performance Determination Block

The PD block may be configured to determine the performance function Fbased on the current inputs x, outputs y, and/or training signal r,denoted by the arrow 404 in FIG. 4. In some implementations, theexternal signal r may comprise the reinforcement signal in thereinforcement learning task. In some implementations, the externalsignal r may comprise reference signal in the supervised learning task.In some implementations, the external signal r comprises the desiredoutput, current costs of control movements, and/or other informationrelated to the current task of the control block (e.g., block 310 inFIG. 3). Depending on the specific learning task (e.g., reinforcement,unsupervised, or supervised) some of the parameters x, y, r may not berequired by the PD block illustrated by the dashed arrows 402_1, 408_1,404_1, respectively, in FIG. 4A The learning apparatus configurationdepicted in FIG. 4 may decouple the PD block from the controller statemodel so that the output of the PD block depends on the learning taskand is independent of the current internal state of the control block.

Generalized Performance Determination

In some implementations the PD block may transmit the external signal rto the learning block (as illustrated by the arrow 404_1) so that:

F(t)=r(t),  (Eqn. 33)

where signal r provides reward and/or punishment signals from theexternal environment. By way of illustration, a mobile robot, controlledby spiking neural network, may be configured to collect resources (e.g.,clean up trash) while avoiding obstacles (e.g., furniture, walls). Inthis example, the signal r may comprise a positive indication (e.g.,representing a reward) at the moment when the robot acquires theresource (e.g., picks up a piece of rubbish) and a negative indication(e.g., representing a punishment) when the robot collides with anobstacle (e.g., wall). Upon receiving the reinforcement signal r, thespiking neural network of the robot controller may change its parameters(e.g., neuron connection weights) in order to maximize the function F(e.g., maximize the reward and minimize the punishment).

In some implementations, the PD block may determine the performancefunction by comparing current system output with the desired outputusing a predetermined measure (e.g., a distance d):

F(t)=d(y(t),y ^(d)(t)),  (Eqn. 34)

where y is the output of the control block (e.g., the block 310 in FIG.3) and r=y^(d) is the external reference signal indicating the desiredoutput that is expected from the control block. In some implementations,the external reference signal r may depend on the input x into thecontrol block. In some implementations, the control apparatus (e.g., theapparatus 300 of FIG. 3) may comprise a spiking neural networkconfigured for pattern classification. A human expert may present to thenetwork an exemplary sensory pattern x and the desired output y^(d) thatdescribes the input pattern x class. The network may change (e.g.,adapt) its parameters w to achieve the desired response on the presentedpairs of input x and desired response y^(d). After learning, the networkmay classify new input stimuli based on one or more past experiences.

In some implementations, such as when characterizing a control blockutilizing analog output signals, the distance function may be determinedusing the squared error estimate as follows:

F(t)=(y(t)−y ^(d)(t))².  (Eqn. 35)

In some implementations, such as those applicable to control blocksusing spiking output signals, the distance measure may be determinedusing the squared error of the convolved signals y, y^(d) as follows:

F=[(y*α)−(y ^(d)*β)]²,  (Eqn. 36)

where α, β are finite impulse response kernels. In some implementations,the distance measure may utilize the mutual information between theoutput signal and the reference signal.

In some implementations, the PD may determine the performance functionby comparing one or more particular characteristic of the output signalwith the desired value of this characteristic:

F=[ƒ(y)−ƒ^(d)(y)]²,  (Eqn. 37)

where ƒ is a function configured to extract the characteristic (orcharacteristics) of interest from the output signal y. By way ofexample, useful with spiking output signals, the characteristic maycorrespond to a firing rate of spikes and the function ƒ(y) maydetermine the mean firing from the output. In some implementations, thedesired characteristic value may be provided through the external signalas

r=ƒ _(d)(y).  (Eqn. 38)

In some implementations, the ƒ^(d)(y) may be calculated internally bythe PD block.

In some implementations, the PD block may determine the performancefunction by calculating the instantaneous mutual information i betweeninputs and outputs of the control block as follows:

F=i(x,y)=−ln(p(y))+ln(p(y|x),  (Eqn. 39)

where p(y) is an unconditioned probability of the current output. It isnoteworthy that the average value of the instantaneous mutualinformation may equal the mutual information I(x,y). This performancefunction may be used to implement ICA (unsupervised learning).

In some implementations, the PD block may determines the performancefunction by calculating the unconditional instantaneous entropy h of theoutput of the control block as follows:

F=h(x,y)=−ln(p(y)).  (Eqn. 40)

where p(y) is an unconditioned probability of the current output. It isnoteworthy that the average value of the instantaneous unconditionalentropy may equal the unconditional H(x,y). This performance functionmay be used to reduce variability in the output of the system foradaptive filtering.

In some implementations, the PD block may determine the performancefunction by calculating the instantaneous Kullback-Leibler divergenced_(KL) between the output probability distribution p(y|x) of the controlblock and some desired probability distribution θ(y|x) as follows:

F=d _(KL)(x,y)=ln(p(y|x))−ln(θ(y|x)).  (Eqn. 41)

It is noteworthy that the average value of the instantaneousKulback-Leibler divergence may equal the d_(KL)(p, θ). The performancefunction of Eqn. 41 may be applied in unsupervised learning tasks inorder to restrict a possible output of the system. For example, if θ(y)is a Poisson distribution of spikes with some firing rate R, thenminimization of this performance function may force the neuron to havethe same firing rate R.

In some implementations, the PD block may determine the performancefunction for the sparse coding. The sparse coding task may be anunsupervised learning task where the adaptive system may discover hiddencomponents in the data that describes data the best with a constraintthat the structure of the hidden components should be sparse:

F=∥x−A(y,w)∥² +∥y∥ ²,  (Eqn. 42)

where the first term quantifies how close the data x can be described bythe current output y, where A(y,w) is a function that describes how todecode an original data from the output. The second term may calculate anorm of the output and may imply restrictions on the output sparseness.

A learning framework of the present innovation may enable generation oflearning rules for a system, which may be configured to solve severalcompletely different tasks-types simultaneously. For example, the systemmay learn to control an actuator while trying to extract independentcomponents from movement trajectories of this actuator. The combinationof tasks may be done as a linear combination of the performancefunctions for each particular problem:

F=C(F ₁ ,F ₂ , . . . , F _(n)),  (Eqn. 43)

where: F₁, F₂, . . . , F_(n) are performance function values fordifferent tasks, and C is a combination function.

In some implementations, the combined cost function C may comprise aweighted linear combination of individual cost functions correspondingto individual learning tasks:

C(F ₁ ,F ₁ , . . . , F ₁)=Σ_(k) a _(k) F _(k),  (Eqn. 44)

where a_(k) are combination weights.

It is recognized by those killed in the arts that linear cost functioncombination described by 44 illustrates one particular implementation ofthe disclosure and other implementations (e.g., a nonlinear combination)may be used as well.

In some implementations, the PD block may be configured to calculate thebaseline of the performance function values (e.g., as a running average)and subtract it from the instantaneous value of the performance functionin order to increase learning speed of learning. The output of the PDblock may comprise a difference between the current value F(t)^(cur) ofthe performance function and its time average

F

:

F(t)=F(t)^(cur) −

F

.  (Eqn. 45)

In some implementations, the time average of the performance functionmay comprise an interval average, where learning occurs over apredetermined interval. A current value of the cost function may bedetermined at individual steps within the interval and may be averagedover all steps. In some implementations, the time average of theperformance function may comprise a running average, where the currentvalue of the cost function may be low-pass filtered according to:

$\begin{matrix}{{\frac{{F(t)}}{t} = {{{- \tau}\; {F(t)}} + {F(t)}^{cur}}},} & \left( {{Eqn}.\mspace{14mu} 46} \right)\end{matrix}$

thereby producing a running average output.

Referring now to FIG. 4A different implementations of the performancedetermination block (e.g., the block 424 of FIG. 4) are shown. The PDblock implementation denoted 434, may be configured to simultaneouslyimplement reinforcement, supervised and unsupervised (RSU) learningrules; and/or receive the input signal x(t) 412, the output signal y(t)418, and/or the learning signal 436. The learning signal 436 maycomprise the reinforcement component r(t) and the desired output(teaching) component y^(d)(t). In one or more implementations, theoutput performance function F_RSU 438 of the RSUPD block may bedetermined in accordance with Eqn. 69 described below.

The PD blocks 444, 445, may implement the reinforcement (R) learningrule. The output 448 of the block 444 may be determined based on theoutput signal y(t) 418 and the reinforcement signal r(t) 446. In one ormore implementations, the output 448 of the RSUPD block may bedetermined in accordance with Eqn. 38. The performance function output449 of the block 445 may be determined based on the input signal x(t),the output signal y(t), and/or the reinforcement signal r(t).

The PD block implementation denoted 454, may be configured to implementsupervised (S) learning rules to generate performance function F_S 458that is dependent on the output signal y(t) value 418 and the teachingsignal y^(d)(t) 456. In one or more implementations, the output 458 ofthe PD 454 block may be determined in accordance with Eqn. 34-Eqn. 37.

The output performance function 468 of the PD block 464 implementingunsupervised learning may be a function of the input x(t) 412 and theoutput y(t) 418. In one or more implementations, the output 468 may bedetermined in accordance with Eqn. 39-Eqn. 42.

The PD block implementation denoted 474 may be configured tosimultaneously implement reinforcement and supervised (RS) learningrules. The PD block 474 may not require the input signal x(t), and mayreceive the output signal y(t) 418 and the teaching signals r(t),y^(d)(t) 476. In one or more implementations, the output performancefunction F_RS 478 of the PD block 474 may be determined in accordancewith Eqn. 43, where the combination coefficient for the unsupervisedlearning is set to zero. By way of example, in some implementationsreinforcement learning task may be to acquire resources by the mobilerobot, where the reinforcement component r(t) provides information aboutacquired resources (reward signal) from the external environment, whileat the same time a human expert shows the robot what should be desiredoutput signal y^(d)(t) to optimally avoid obstacles. By setting a highercoefficient to the supervised part of the performance function, therobot may be trained to try to acquire the resources if it does notcontradict with human expert signal for avoiding obstacles.

The PD block implementation denoted 475 may be configured tosimultaneously implement reinforcement and supervised (RS) learningrules. The PD block 475 output may be determined based the output signal418, the learning signals 476, comprising the reinforcement componentr(t) and the desired output (teaching) component y^(d)(t) and on theinput signal 412, that determines the context for switching betweensupervised and reinforcement task functions. By way of example, in someimplementations, reinforcement learning task may be used to acquireresources by the mobile robot, where the reinforcement component r(t)provides information about acquired resources (reward signal) from theexternal environment, while at the same time a human expert shows therobot what should be desired output signal y^(d)(t) to optimally avoidobstacles. By recognizing obstacles, avoidance context on the basis ofsome clues in the input signal, the performance signal may be switchedbetween supervised and reinforcement. That may allow the robot to betrained to try to acquire the resources if it does not contradict withhuman expert signal for avoiding obstacles. In one or moreimplementations, the output performance function 479 of the PD 475 blockmay be determined in accordance with Eqn. 43, where the combinationcoefficient for the unsupervised learning is set to zero.

The PD block implementation denoted 484 may be configured tosimultaneously implement reinforcement, and unsupervised (RU) learningrules. The output 488 of the block 484 may be determined based on theinput and output signals 412, 418, in one or more implementations, inaccordance with Eqn. 43. By way of example, in some implementations ofsparse coding (unsupervised learning), the task of the adaptive systemon the robot may be not only to extract sparse hidden components fromthe input signal, but to pay more attention to the components that arebehaviorally important for the robot (that provides more reinforcementafter they can be used).

The PD block implementation denoted 494, which may be configured tosimultaneously implement supervised and unsupervised (SU) learningrules, may receive the input signal x(t) 412, the output signal y(t)418, and/or the teaching signal y^(d)(t) 436. In one or moreimplementations, the output performance function F_SU 438 of the SU PDblock may be determined in accordance with Eqn. 68 described below.

By the way of example, the stochastic learning system (that isassociated with the PD block implementation 494) may be configured tolearn to implement unsupervised data categorization (e.g., using sparsecoding performance function), while simultaneously receiving externalsignal that is related to the correct category of particular inputsignals. In one or more implementations such reward signal may beprovided by a human expert.

Performance Determination for Spiking Neurons

In one or more implementations of reinforcement learning, the PD block(e.g., the block 424 of FIG. 4) may generate the performance signalbased on analog and/or spiking reward signal r (e.g., the signal 404 ofFIG. 4). In one implementation, the performance signal F (e.g., thesignal 428 of FIG. 4) may comprise the reward signal r(t), transmittedto the PA block (e.g., the block 426 of FIG. 4) by the PD block.

In one or more implementations related to analog reward signal, in orderto reduce computational load on the PA block related to application ofweight changes, the PD block may transform the analog reward r(t) intospike form.

In one or more implementations of supervised learning, the currentperformance F may be determined based on the output of the neuron andthe external reference signal (e.g., the desired output y^(d)(t)). Forexample, a distance measure may be calculated using a low-pass filteredversion of the desired y^(d)(t) and actual y(t) outputs. In someimplementations, a running distance between the filtered spike trainsmay be determined according to

$\begin{matrix}{{{F\left( {{x(t)},{y(t)}} \right)} = {{\left( {{\int_{- \infty}^{t}{{y(s)}{a\left( {\tau - s} \right)}{\tau}}} - {\int_{- \infty}^{t}{{y^{d}(s)}{b\left( {\tau - s} \right)}{\tau}}}} \right)^{2}\mspace{14mu} {where}\text{:}\mspace{14mu} {y(t)}} = {\sum\limits_{i}{\delta \left( {t - t_{i}^{out}} \right)}}}},{{y^{d}(t)} = {\sum\limits_{j}{\delta \left( {t - t_{j}^{d}} \right)}}},} & \left( {{Eqn}.\mspace{14mu} 47} \right)\end{matrix}$

with y(t) and y^(d) (t) being the actual and desired output spiketrains; δ(t) is the Dirac delta function; t_(i) ^(out), t_(j) ^(d) arethe output and desired spike times, respectively; and a(t), b(t) arepositive finite-response kernels. In some implementations, the kernela(t) may comprise an exponential trace: a(t)=e^(−t/τ) ^(a) .

In some implementations of supervised learning, spiking neuronal networkmay be configured to learns to minimize a Kullback-Leibler distancebetween the actual and desired output:

F(x(t),y(t))=D _(KL)(y(t)∥r(t)).  (Eqn. 48)

In some implementations, if r(t) is a Poisson spike train with a fixedfiring rate, the D_(KL) learning may enable stabilization of theneuronal firing rate.

In some implementations of supervised learning, for example in“information bottleneck”, part of the performance optimization maycomprise maximization of the mutual information between the actualoutput y(t) and some reference signal r(t). For a given input andoutput, the performance function may be expressed as:

F(x(t),y(t))=I(y(t),r(t)).  (Eqn. 49)

In one or more implementations of unsupervised learning, the costfunction may be obtained by a minimization of the conditionalinformational entropy of the output spiking pattern:

F(x,y)=H(y|x)  (Eqn. 50)

so as to provide a more stable neuron output y for a given input x.

Parameter Changing Block

The parameter changing PA block (the block 426 in FIG. 4) may determinechanges of the control block parameters Δw_(i) according to apredetermined learning algorithm, based on the performance function Fand the gradient g it receives from the PD block 424 and the GD block422, as indicated by the arrows marked 428, 430, respectively, in FIG.4. Particular implementation of the learning algorithm within the block426 may depend on the type of the learning task (e.g., online or batchlearning) used by the learning block 320 of FIG. 3.

Several exemplary implementations of PA learning algorithms applicablewith spiking control signals are described below. In someimplementations, the PA learning algorithms may comprise amultiplicative online learning rule, where control parameter changes aredetermined as follows:

Δ

(t)=γF(t)

(t),  (Eqn. 51)

where γ is the learning rate configured to determine speed of learningadaptation. The learning method implementation according to (Eqn. 51)may be advantageous in applications where the performance function F(t)depends on the current values of the inputs x, outputs y, and/or signalr.

In some implementations, the control parameter adjustment Δw may bedetermined using an accumulation of the score function gradient and theperformance function values, and applying the changes at a predeterminedtime instance (corresponding to, e.g., the end of the learning epoch):

$\begin{matrix}{{{\Delta \; {\overset{r}{w}(t)}} = {\frac{\gamma}{N^{2}} \cdot {\sum\limits_{i = 0}^{N - 1}{{F\left( {t - {i\; \Delta \; t}} \right)} \cdot {\sum\limits_{i = 0}^{N - 1}{\overset{r}{g}\left( {t - {i\; \Delta \; t}} \right)}}}}}},} & \left( {{Eqn}.\mspace{14mu} 52} \right)\end{matrix}$

where: T is a finite interval over which the summation occurs; N is thenumber of steps; and Δt is the time step determined as T/N.The summation interval T in Eqn. 52 may be configured based on thespecific requirements of the control application. By way ofillustration, in a control application where a robotic arm is configuredto reaching for an object, the interval may correspond to a time fromthe start position of the arm to the reaching point and, in someimplementations, may be about 1 s-50 s. In a speech recognitionapplication, the time interval T may match the time required topronounce the word being recognized (typically less than 1 s-2 s). Insome implementations of spiking neuronal networks, Δt may be configuredin range between 1 ms and 20 ms, corresponding to 50 steps (N=50) in onesecond interval.

The method of Eqn. 52 may be computationally expensive and may notprovide timely updates. Hence, it may be referred to as the non-local intime due to the summation over the interval T. However, it may lead tounbiased estimation of the gradient of the performance function.

In some implementations, the control parameter adjustment Δw_(i) may bedetermined by calculating the traces of the score function e_(i)(t) forindividual parameters w_(i). In some implementations, the traces may becomputed using a convolution with an exponential kernel β as follows:

{right arrow over (e)}(t+Δt)=β{right arrow over (e)}(t)+{right arrowover (g)}(t),  (Eqn. 53)

where β is the decay coefficient. In some implementations, the tracesmay be determined using differential equations:

$\begin{matrix}{{\frac{}{t}{\overset{->}{e}(t)}} = {{{- \tau}\; {\overset{->}{e}(t)}} + {{\overset{->}{g}(t)}.}}} & \left( {{Eqn}.\mspace{14mu} 54} \right)\end{matrix}$

The control parameter w may then be adjusted as:

{right arrow over (Δw)}(t)=γF(t){right arrow over (e)}(t),  (Eqn. 55)

where γ is the learning rate. The method of Eqn. 53-Eqn. 55 may beappropriate when a performance function depends on current and pastvalues of the inputs and outputs and may be referred to as the OLPOMDPalgorithm. While it may be local in time and computationally simple, itmay lead to biased estimate of the performance function. By way ofillustration, the methodology described by Eqn. 53-Eqn. 55 may be used,in some implementations, in a rescue robotic device configured to locateresources (e.g., survivors, or unexploded ordinance) in a building. Theinput x may correspond to the robot current position in the building.The reward r (e.g., the successful location events) may depend on thehistory of inputs and on the history of actions taken by the agent(e.g., left/right turns, up/down movement, etc.).

In some implementations, the control parameter adjustment Δw determinedusing methodologies of the Eqns. 16, 17, 19 may be further modifiedusing, in one variant, gradient with momentum according to:

Δ

(t)

μΔ

(t−Δt)+Δ

(t),  (Eqn. 56)

where μ is the momentum coefficient. In some implementations, the signof gradient may be used to perform learning adjustments as follows:

$\begin{matrix}\left. {\Delta \; {w_{i}(t)}}\Rightarrow{\frac{\Delta \; {w_{i}(t)}}{{\Delta \; {w_{i}(t)}}}.} \right. & \left( {{Eqn}.\mspace{14mu} 57} \right)\end{matrix}$

In some implementations, gradient descend methodology may be used forlearning coefficient adaptation.

In some implementations, the gradient signal g, determined by the PDblock 422 of FIG. 4, may be subsequently modified according to anothergradient algorithm, as described in detail below. In someimplementations, these modifications may comprise determining naturalgradient, as follows:

$\begin{matrix}{{{\Delta \; \overset{r}{w}} = {{\langle{\overset{r}{g} \cdot \overset{r_{T}}{g}}\rangle}_{x,y}^{- 1} \cdot {\langle{\overset{r}{g} \cdot F}\rangle}_{x,y}}},} & \left( {{Eqn}.\mspace{14mu} 58} \right)\end{matrix}$

where ({right arrow over (g)}{right arrow over (g)}^(T))_(x,y) is theFisher information metric matrix. Applying the following transformationto Eqn. 21:

$\begin{matrix}{{{\langle{\overset{r}{g}\left( {{\overset{r_{T}}{g}\Delta \; \overset{r}{w}} - F} \right)}\rangle}_{x,y} = 0},} & \left( {{Eqn}.\mspace{14mu} 59} \right)\end{matrix}$

natural gradient from linear regression task may be obtained as follows:

GΔ{right arrow over (w)}={right arrow over (F)}  (Eqn. 60)

where G=[{right arrow over (g₀ ^(T))}, . . . , {right arrow over (g_(n)^(T))}] is a matrix comprising n samples of the score function g, {rightarrow over (F^(T))}=[F₀, . . . , F_(n)] is the a vector of performancefunction samples, and n is a number of samples that should be equal orgreater of the number of the parameters w_(i). While the methodology ofEqn. 58-Eqn. 60 may be computationally expensive, it may help dealingwith ‘plateau’-like landscapes of the performance function.

Signal Processing Apparatus

In one or more implementations, the generalized learning frameworkdescribed supra may enable implementing signal processing blocks withtunable parameters w. Using the learning block framework that providesanalytical description of individual types of signal processing blockmay enable it to automatically calculate the appropriate score function

$\frac{\partial{h\left( {xy} \right)}}{\partial w_{i}}$

for individual parameters of the block. Using the learning architecturedescribed in FIG. 3, a generalized implementation of the learning blockmay enable automatic changes of learning parameters w by individualblocks based on high level information about the subtask for each block.A signal processing system comprising one or more of such generalizedlearning blocks may be capable of solving different learning tasksuseful in a variety of applications without substantial intervention ofthe user. In some implementations, such generalized learning blocks maybe configured to implement generalized learning framework describedabove with respect to FIGS. 3-4A and delivered to users. In developingcomplex signal processing systems, the user may connect differentblocks, and/or specify a performance function and/or a learningalgorithm for individual blocks. This may be done, for example, with thespecial graphical user interface (GUI), which may allow blocks to beconnected using a mouse or other input peripheral by clicking onindividual blocks and using defaults or choosing the performancefunction and a learning algorithm from a predefined list. Users may notneed to re-create a learning adaptation framework and may rely on theadaptive properties of the generalized learning blocks that adapt to theparticular learning task. When the user desires to add a new type ofblock into the system, he may need to describe it in a way suitable toautomatically calculate a score functions for individual parameters.

FIG. 5 illustrates one exemplary implementation of a robotic apparatus500 comprising adaptive controller apparatus 512. In someimplementations, the adaptive controller 520 may be configured similarto the apparatus 300 of FIG. 3 and may comprise generalized learningblock (e.g., the block 420), configured, for example according to theframework described above with respect to FIG. 4, supra, is shown anddescribed. The robotic apparatus 500 may comprise the plant 514,corresponding, for example, to a sensor block and a motor block (notshown). The plant 514 may provide sensory input 502, which may include astream of raw sensor data (e.g., proximity, inertial, terrain imaging,and/or other raw sensor data) and/or preprocessed data (e.g., velocity,extracted from accelerometers, distance to obstacle, positions, and/orother preprocessed data) to the controller apparatus 520. The learningblock of the controller 520 may be configured to implement reinforcementlearning, according to, in some implementations Eqn. 38, based on thesensor input 502 and reinforcement signal 504 (e.g., obstacle collisionsignal from robot bumpers, distance from robotic arm endpoint to thedesired position), and may provide motor commands 506 to the plant. Thelearning block of the adaptive controller apparatus (e.g., the apparatus520 of FIG. 5) may perform learning parameter (e.g., weight) adaptationusing reinforcement learning approach without having any priorinformation about the model of the controlled plant (e.g., the plant 514of FIG. 5). The reinforcement signal r(t) may inform the adaptivecontroller that the previous behavior led to “desired” or “undesired”results, corresponding to positive and negative reinforcements,respectively. While the plant 514 must be controllable (e.g., via themotor commands in FIG. 5) and the control system may be required to haveaccess to appropriate sensory information (e.g., the data 502 in FIG.5), detailed knowledge of motor actuator dynamics or of structure andsignificance of sensory signals may not be required to be known by thecontroller apparatus 520.

It will be appreciated by those skilled in the arts that thereinforcement learning configuration of the generalized learningcontroller apparatus 520 of FIG. 5 is used to illustrate one exemplaryimplementation of the disclosure and myriad other configurations may beused with the generalized learning framework described herein. By way ofexample, the adaptive controller 520 of FIG. 5 may be configured for:(i) unsupervised learning for performing target recognition, asillustrated by the adaptive controller 520_3 of FIG. 5A, receivingsensory input and output signals (x,y) 522_3; (ii) supervised learningfor performing data regression, as illustrated by the adaptivecontroller 520_3 receiving output signal 522_1 and teaching signal 504_1of FIG. 5A; and/or (iii) simultaneous supervised and unsupervisedlearning for performing platform stabilization, as illustrated by theadaptive controller 520_2 of FIG. 5A, receiving input 522_2 and learning504_2 signals.

FIGS. 5B-5C illustrate dynamic tasking by a user of the adaptivecontroller apparatus (e.g., the apparatus 320 of FIG. 3A or 520 of FIG.5, described supra) in accordance with one or more implementations.

A user of the adaptive controller 520_4 of FIG. 5B may utilize a userinterface (textual, graphics, touch screen, etc.) in order to configurethe task composition of the adaptive controller 520_4, as illustrated bythe example of FIG. 5B. By way of illustration, at one instance for oneapplication the adaptive controller 520_4 of FIG. 5B may be configuredto perform the following tasks: (i) task 550_1 comprising sensorycompressing via unsupervised learning; (ii) task 550_2 comprising rewardsignal prediction by a critic block via supervised learning; and (ii)task 550_3 comprising implementation of optimal action by an actor blockvia reinforcement learning. The user may specify that task 550_1 mayreceive external input {X} 542, comprising, for example raw audio orvideo stream, output 546 of the task 550_1 may be routed to each oftasks 550_2, 550_3, output 547 of the task 550_2 may be routed to thetask 550_3; and the external signal {r} (544) may be provided to each oftasks 550_2, 550_3, via pathways 544_1, 544_2, respectively asillustrated in FIG. 5B. In the implementation illustrated in FIG. 5B,the external signal {r} may be configured as {r}={y^(d)(t), r(t)}, thepathway 544_1 may carry the desired output y^(d)(t), while the pathway544_2 may carry the reinforcement signal r(t).

Once the user specifies the learning type(s) associated with each task(unsupervised, supervised and reinforcement, respectively) thecontroller 520_4 of FIG. 5B may automatically configure the respectiveperformance functions, without further user intervention. By way ofillustration, performance function F_(u) of the task 550_1 may bedetermined based on (i) ‘sparse coding’; and/or (ii) maximization ofinformation. Performance function F_(s) of the task 550_2 may bedetermined based on minimizing distance between the actual output 547(prediction pr) d(r, pr) and the external reward signal r 544_1.Performance function F_(r) of the task 550_3 may be determined based onmaximizing the difference F=r−pr. In some implementations, the end usermay select performance functions from a predefined set and/or the usermay implement a custom task.

At another instance in a different application, illustrated in FIG. 5C,the controller 520_4 may be configured to perform a different set oftask: (i) the task 550_1, described above with respect to FIG. 5B; andtask 552_4, comprising pattern classification via supervised learning.As shown in FIG. 5C, the output of task 550_1 may be provided as theinput 566 to the task 550_4.

Similarly to the implementation of FIG. 5B, once the user specifies thelearning type(s) associated with each task (unsupervised and supervised,respectively) the controller 520_4 of FIG. 5C may automaticallyconfigure the respective performance functions, without further userintervention. By way of illustration, the performance functioncorresponding to the task 550_4 may be configured to minimize distancebetween the actual task output 568 (e.g., a class {Y} to which a sensorypattern belongs) and human expert supervised signal 564 (the correctclass y^(d)).

Generalized learning methodology described herein may enable thelearning apparatus 520_4 to implement different adaptive tasks, by, forexample, executing different instances of the generalized learningmethod, individual ones configured in accordance with the particulartask (e.g., tasks 550_1, 550_2, 550_3, in FIG. 5B, and 550_4, 550_5 inFIG. 5C). The user of the apparatus may not be required to knowimplementation details of the adaptive controller (e.g., specificperformance function selection, and/or gradient determination). Instead,the user may ‘task’ the system in terms of task functions andconnectivity.

Partitioned Network Apparatus

FIGS. 6A-6B illustrate exemplary implementations of reconfigurablepartitioned neural network apparatus comprising generalized learningframework, described above. The network 600 of FIG. 6A may compriseseveral partitions 610, 620, 630, comprising one or more of nodes 602receiving inputs 612 {X} via connections 604, and providing outputs viaconnections 608.

In one or more implementations, the nodes 602 of the network 600 maycomprise spiking neurons (e.g., the neurons 730 of FIG. 9, describedbelow), the connections 604, 608 may be configured to carry spikinginput into neurons, and spiking output from the neurons, respectively.The neurons 602 may be configured to generate responses (as describedin, for example, U.S. patent application Ser. No. 13/152,105 filed onJun. 2, 2011, and entitled “APPARATUS AND METHODS FOR TEMPORALLYPROXIMATE OBJECT RECOGNITION”, incorporated by reference herein in itsentirety) which may be propagated via feed-forward connections 608.

In some implementations, the network 600 may comprise artificialneurons, such as for example, spiking neurons described by U.S. patentapplication Ser. No. 13/152,105 filed on Jun. 2, 2011, and entitled“APPARATUS AND METHODS FOR TEMPORALLY PROXIMATE OBJECT RECOGNITION”,incorporated supra, artificial neurons with sigmoidal activationfunction, binary neurons (perceptron), radial basis function units,and/or fuzzy logic networks.

Different partitions of the network 600 may be configured, in someimplementations, to perform specialized functionality. By way ofexample, the partition 610 may adapt raw sensory input of a roboticapparatus to internal format of the network (e.g., convert analog signalrepresentation to spiking) using for example, methodology described inU.S. patent application Ser. No. 13/314,066, filed Dec. 7, 2001,entitled “NEURAL NETWORK APPARATUS AND METHODS FOR SIGNAL CONVERSION”,incorporated herein by reference in its entirety. The output {Y1} of thepartition 610 may be forwarded to other partitions, for example,partitions 620, 630, as illustrated by the broken line arrows 618, 618_1in FIG. 6A. The partition 620 may implement visual object recognitionlearning that may require training input signal y^(d) _(j)(t) 616, suchas for example an object template and/or a class designation(friend/foe). The output {Y2}) of the partition 620 may be forwarded toanother partition (e.g., partition 630) as illustrated by the dashedline arrow 628 in FIG. 6A. The partition 630 may implement motor controlcommands required for the robotic arm to reach and grasp the identifiedobject, or motor commands configured to move robot or camera to a newlocation, which may require reinforcement signal r(t) 614. The partition630 may generate the output {Y} 638 of the network 600 implementingadaptive controller apparatus (e.g., the apparatus 520 of FIG. 5). Thehomogeneous configuration of the network 600, illustrated in FIG. 6A,may enable a single network comprising several generalized nodes of thesame type to implement different learning tasks (e.g., reinforcement andsupervised) simultaneously.

In one or more implementations, the input 612 may comprise input fromone or more sensor sources (e.g., optical input {Xopt} and audio input{Xaud}) with each modality data being routed to the appropriate networkpartition, for example, to partitions 610, 630 of FIG. 6A, respectively.

The homogeneous nature of the network 600 may enable dynamicreconfiguration of the network during its operation. FIG. 6B illustratesone exemplary implementation of network reconfiguration in accordancewith the disclosure. The network 640 may comprise partition 650, whichmay be configured to perform unsupervised learning task, and partition660, which may be configured to implement supervised and reinforcementlearning simultaneously. The network configuration of FIG. 6B may beused to perform signal separation tasks by the partition 650 and signalclassification tasks by the partition 660. The partition 650 may beoperated according to unsupervised learning rule and may generate output{Y3} denoted by the arrow 658 in FIG. 6B. The partition 660 may beoperated according to a combined reinforcement and supervised rule, mayreceive supervised and reinforcement input 656, and/or may generate theoutput {Y4} 668.

The dynamic network learning reconfiguration illustrated in FIGS. 6A-6Bmay be used, for example, in an autonomous robotic apparatus performingexploration tasks (e.g., a pipeline inspection autonomous underwatervehicle (AUV), or space rover, explosive detection, and/or mineexploration). When certain functionality of the robot is not required(e.g., the arm manipulation function) the available network resources(i.e., the nodes 602) may be reassigned to perform different tasks. Suchreuse of network resources may be traded for (i) smaller networkprocessing apparatus, having lower cost, size and consuming less power,as compared to a fixed pre-determined configuration; and/or (ii)increased processing capability for the same network capacity.

As is appreciated by those skilled in the arts, the reconfigurationmethodology described supra may comprise a static reconfiguration, whereparticular node populations are designated in advance for specificpartitions (tasks); a dynamic reconfiguration, where node partitions aredetermined adaptively based on the input information received by thenetwork and network state; and/or a semi-static reconfiguration, wherestatic partitions are assigned predetermined life-span.

Spiking Network Apparatus

Referring now to FIG. 7, one implementation of spiking network apparatusfor effectuating the generalized learning framework of the disclosure isshown and described in detail. The network 700 may comprise at least onestochastic spiking neuron 730, operable according to, for example, aSpike Response Model, and configured to receive n-dimensional inputspiking stream X(t) 702 via n-input connections 714. In someimplementations, the n-dimensional spike stream may correspond ton-input synaptic connections into the neuron. As shown in FIG. 7,individual input connections may be characterized by a connectionparameter 712 w_(ij) that is configured to be adjusted during learning.In one or more implementation, the connection parameter may compriseconnection efficacy (e.g., weight). In some implementations, theparameter 712 may comprise synaptic delay. In some implementations, theparameter 712 may comprise probabilities of synaptic transmission.

The following signal notation may be used in describing operation of thenetwork 700, below:

${y(t)} = {\sum\limits_{i}^{\;}\; {\delta \left( {t - t_{i}} \right)}}$

denotes the output spike pattern, corresponding to the output signal 708produced by the control block 710 of FIG. 3, where t_(i) denotes thetimes of the output spikes generated by the neuron;

${y^{d}(t)} = {\sum\limits_{t_{i}}^{\;}\; {\delta \left( {t - t_{i}^{d}} \right)}}$

denotes the teaching spike pattern, corresponding to the desired (orreference) signal that is part of external signal 404 of FIG. 4, wheret_(i) ^(d) denotes the times when the spikes of the reference signal arereceived by the neuron;

${{y^{+}(t)} = {\sum\limits_{t_{i}}^{\;}\; {\delta \left( {t - t_{i}^{+}} \right)}}};$${y^{-}(t)} = {\sum\limits_{t_{i}}^{\;}\; {\delta \left( {t - t_{i}^{-}} \right)}}$

denotes the reinforcement signal spike stream, corresponding to signal304 of FIG. 3. and external signal 404 of FIG. 4, where t_(i) ⁺, t_(i) ⁻denote the spike times associated with positive and negativereinforcement, respectively.

In some implementations, the neuron 730 may be configured to receivetraining inputs, comprising the desired output (reference signal)y^(d)(t) via the connection 704. In some implementations, the neuron 730may be configured to receive positive and negative reinforcement signalsvia the connection 704.

The neuron 730 may be configured to implement the control block 710(that performs functionality of the control block 310 of FIG. 3) and thelearning block 720 (that performs functionality of the control block 320of FIG. 3, described supra.) The block 710 may be configured to receiveinput spike trains X(t), as indicated by solid arrows 716 in FIG. 7, andto generate output spike train y(t) 708 according to a Spike ResponseModel neuron which voltage v(t) is calculated as:

${{v(t)} = {\sum\limits_{i,k}^{\;}\; {w_{i} \cdot {\alpha \left( {t - t_{i}^{k}} \right)}}}},$

where w_(i) w_(i) represents weights of the input channels, t_(i) ^(k)represents input spike times, and α(t)=(t/τ_(α))e^(1-(t/τ) ^(α) ⁾represents an alpha function of postsynaptic response, where τ_(α)represents time constant (e.g., 3 ms and/or other times). Aprobabilistic part of a neuron may be introduced using the exponentialprobabilistic threshold. Instantaneous probability of firing λ(t) may becalculated as λ(t)=e^((v(t)-Th)κ), where Th—represents a thresholdvalue, and κ represents stochasticity parameter within the controlblock. State variables g (probability of firing λ(t) for this system)associated with the control model may be provided to the learning block720 via the pathway 705. The learning block 720 of the neuron 730 mayreceive the output spike train y(t) via the pathway 708_1. In one ormore implementations (e.g., unsupervised or reinforcement learning), thelearning block 720 may receive the input spike train (not shown). In oneor more implementations (e.g., supervised or reinforcement learning) thelearning block 720 may receive the learning signal, indicated by dashedarrow 704_1 in FIG. 7. The learning block determines adjustment of thelearning parameters w, in accordance with any methodologies describedherein, thereby enabling the neuron 730 to adjust, inter alia,parameters 712 of the connections 714.

Exemplary Methods Generalized Learning Rules

Referring now to FIG. 8A one exemplary implementation of the generalizedlearning method of the disclosure for use with, for example, thelearning block 420 of FIG. 4, is described in detail. The method 800 ofFIG. 8A may allow the learning apparatus to: (i) implement differentlearning rules (e.g., supervised, unsupervised, reinforcement, and/orother learning rules); and (ii) simultaneously support more than onerule (e.g., combination of supervised, unsupervised, reinforcement rulesdescribed, for example by Eqn. 43) using the same hardware/softwareconfiguration.

At step 802 of method 800 the input information may be received. In someimplementations (e.g., unsupervised learning) the input information maycomprise the input signal x(t), which may comprise raw or processedsensory input, input from the user, and/or input from another part ofthe adaptive system. In one or more implementations, the inputinformation received at step 802 may comprise learning task identifierconfigured to indicate the learning rule configuration (e.g., Eqn. 43)that should be implemented by the learning block. In someimplementations, the indicator may comprise a software flag transitedusing a designated field in the control data packet. In someimplementations, the indicator may comprise a switch (e.g., effectuatedvia a software commands, a hardware pin combination, or memoryregister).

At step 804, learning framework of the performance determination block(e.g., the block 424 of FIG. 4) may be configured in accordance with thetask indicator. In one or more implementations, the learning structuremay comprise, inter alia, performance function configured according toEqn. 43. In some implementations, parameters of the control block, e.g.,number of neurons in the network, may be configured as well.

At step 808, the status of the learning indicator may be checked todetermine whether additional learning input is required. In someimplementations, the additional learning input may comprisereinforcement signal r(t). In some implementations, the additionallearning input may comprise desired output (teaching signal) y^(d)(t),described above with respect to FIG. 4.

If required, the external learning input may be received by the learningblock at step 808.

At step 812, the value of the present performance may be computedperformance function F(x,y,r) configured at the prior step. It will beappreciated by those skilled in the arts, that when performance functionis evaluated for the first time (according, for example to Eqn. 35) andthe controller output y(t) is not available, a pre-defined initial valueof y(t) (e.g., zero) may be used instead.

At step 814, gradient g(t) of the score function (logarithm of theconditional probability of output) may be determined by the GD block(e.g., The block 422 of FIG. 4).

At step 816, learning parameter w update may be determined by theParameter Adjustment block (e.g., block 426 of FIG. 4) using theperformance function F and the gradient g, determined at steps 812, 814,respectively. In some implementations, the learning parameter update maybe implemented according to Eqns. 22-31.

At step 814, gradient g(t) of the score function may be determinedaccording, by the GD block (e.g., block 422 of FIG. 4). The learningparameter update may be subsequently provided to the control block(e.g., block 310 of FIG. 3).

At step 818, the control output y(t) of the controller may be updatedusing the input signal x(t) (received via the pathway 820) and theupdated learning parameter Aw.

FIG. 8B illustrates a method of dynamic controller reconfiguration basedon learning tasks, in accordance with one or more implementations.

At step 822 of method 830, the input information may be received. Asdescribed above with respect to FIG. 8A, in some implementations, theinput information may comprise the input signal x(t) and/or learningtask identifier configured to indicate the learning rule configuration(e.g., Eqn. 43) that should be implemented buy the learning block.

At step 834, the controller partitions (e.g., the partitions 520_6,520_7, 520_8, 520_9, of FIG. 5B, and/or partitions 610, 620, 630 of FIG.6A) may be configured in accordance with the learning rules (e.g.,supervised, unsupervised, reinforcement, and/or other learning rules)corresponding to the task received at step 832. Subsequently, individualpartitions may be operated according to, for example, the method 800described with respect to FIG. 8A.

At step 836, a check may be performed as to whether the new task (ortask assortment) is received. If no new tasks are received, the methodmay proceed to step 834. If new tasks are received that requirecontroller repartitioning, such as for example, when exploration roboticdevice may need to perform visual recognition tasks when stationary, themethod may proceed to step 838.

At step 838, current partition configuration (e.g., input parameter,state variables, neuronal composition, connection map, learningparameter values and/or rules, and/or other information associated withthe current partition configuration) may be saved in a nonvolatilememory.

At step 840, the controller state and partition configurations may resetand the method proceeds to step 832, where a new partition set may beconfigured in accordance with the new task assortment received at step836. Method 800 of FIG. 8B may enable, inter alia, dynamic partitionreconfiguration as illustrated in FIGS. 5B, 6A-6B, supra.

Automatic Derivation of Eligibility Traces

FIG. 18 illustrates exemplary data flow of automatic determination ofeligibility traces for use with spiking neuron networks (e.g., thenetwork 600 of FIG. 6A), in accordance with one or more implementations.

At step 1802 of method 1800, the state vector q, describing dynamicmodel of the neuron, may be provided. As described above with respect toEqn. 7, the state vector may comprise membrane voltage and/or currentand may be provided as user input, in some implementations.

At step 1804, the dynamic state of the neuron is described, such as, forexample the structure of IF model or IZ neuron model. In one or moreimplementations, neuron state dynamics may be specified in generalizedsymbolic form as:

V(q)^(T)=(V ₁(q),V ₂(q), . . . , V _(n)(q)),

R(q)^(T)=(R ₁(q),R ₂(q), . . . , R _(n)(q)),

G(q)^(T)=(G ₁(q),G ₂(q), . . . , G _(n)(q)),q ^(T) ={q ₀ , . . . , q_(n)}.  (Eqn. 61)

At step 1806, partial derivatives of the state functions of Eqn. 61 maybe determined as:

$\begin{matrix}{\frac{\partial V_{i}}{\partial q_{j}},\frac{\partial R_{i}}{\partial q_{j}},{\frac{\partial G_{i}}{\partial q_{j}}.}} & \left( {{Eqn}.\mspace{14mu} 62} \right)\end{matrix}$

At step 1808, Jacobian matrices J_(V)(q), J_(R)(q), J_(c)(q), associatedwith the respective dynamic neuronal model may be constructed. In someimplementations of the IF neurons, the Jacobian matrices may bedetermined according to Eqn. 23. In one or more implementations of theIZ neuronal neurons, the Jacobian matrices may be determined accordingto Eqn. 26.

At step 1818, state traces SP), associated with the respective dynamicneuronal model, may be determined.

At step 1810, instantaneous probability density (IPD) λ(q(t)) of theneuron may be constructed. In some implementations of the IF neuronsand/or IZ neuronal neurons, the IPD may be determined according to Eqn.2-Eqn. 4.

At step 1812, partial derivatives

$\left( \frac{\partial\lambda}{\partial\overset{\_}{q}} \right)^{T}$

of the IPD with respect to the state vector q may be determined.

At step 1820, the instantaneous PDF derivative

$\frac{\partial\lambda}{\partial w_{i}}$

may be determined using the partial derivatives of IPD from the step1812 and the gradient from step 1818. In one or more implementations,the instantaneous PDF derivative may be determined using Eqn. 19.

At step 1822, the score function

${gi} = \frac{\partial h_{\Delta \; t}}{\partial w_{i}}$

may be determined using, for example, Eqn. 17. In one or moreimplementations of IF neuronal model, the exponential stochasticthreshold may be implemented using Eqn. 2 and the score function g_(i)may be determined using Eqn. 32.

In some implementations of IZ neuronal model, the exponential stochasticthreshold may be implemented using Eqn. 28 and the score function g, maybe determined using Eqn. 17 and Eqn. 29.

FIG. 19A presents one exemplary implementation of Python scriptconfigured to effectuate automatic derivation of eligibility traces ofthe method 1800 of FIG. 18. In FIG. 19A, the designators #18XX refer tothe respective steps of the method 1800 of FIG. 18, according to one ormore implementations.

FIG. 19B is a python script illustrates exemplary object constructs foruse with the python script of FIG. 19A. The script shown in FIG. 19B isconfigured to interface with MATLAB® symbolic computations engine,according to one implementation. It will be appreciated by those skilledin the arts, that various other symbolic computations computer aideddesign (CAD) tools (e.g., Mathematica, etc.) may be used with themethodology described with respect to FIGS. 18-19B.

Performance Results

FIGS. 9A through 17B present performance results obtained duringsimulation and testing by the Assignee hereof, of exemplary computerizedspiking network apparatus configured to implement generalized learningframework described above with respect to FIGS. 3-6B. The exemplaryapparatus, in one implementation, comprises learning block (e.g., theblock 420 of FIG. 4) that implemented using spiking neuronal network700, described in detail with respect to FIG. 7, supra.

The average performance (e.g. the function

F

_(x,y,r) average of Eqn. 33-Eqn. 43) may be determined over a timeinterval Tav that is configured in accordance with the specificapplication. In some implementations, the spike rate of the networkoutput y(t) may be configured between 5 and 100 Hz. In one or moreimplementations, the Tav may be configured to exceed the spike rate ofoutput by a factor of 5 to 10000. In one such implementation, the spikerate may comprise 70 Hz output, and the averaging time may be selectedat about 100 s.

Combined Supervised and Reinforcement Learning Tasks

In some implementations, in accordance with the framework described by,inter alia, Eqn. 43, the cost function F_(sr), corresponding to acombination of supervised and reinforcement learning tasks, may beexpressed as follows:

F _(sr) =aF _(sup) +bF _(reinf),

where F_(sup) and F_(reinf) are the cost functions for the supervisedand reinforcement learning tasks, respectively, and a, b arecoefficients determining relative contribution of each cost component tothe combined cost function. By varying the coefficients a, b duringdifferent simulation runs of the spiking network, effects of relativecontribution of each learning method on the network learning performancemay be investigated.

In some implementations, such as those involving classification ofspiking input patterns derived from speech data in order to determinespeaker identity, the supervised learning cost function may comprise aproduct of the desired spiking pattern y^(d) (t) (belonging to aparticular speaker) with filtered output spike train y(t). In someimplementations, such as those involving a low pass exponential filterkernel, the F_(sup) may be computed using the following expression:

$\begin{matrix}{{F_{\sup} = {\int_{- \infty}^{t}{\left( {{y(s)}^{{- {({t - s})}}/\tau_{d}}\ {s}} \right)\left( {{y^{d}(t)} - C} \right)}}},} & \left( {{Eqn}.\mspace{14mu} 63} \right)\end{matrix}$

where τ_(d) is the trace decay constant, and C is the bias constantconfigured to introduce penalty associated with extra activity of theneuron does not corresponding to the desired spike train.

The cost function for reinforcement learning may be determined as a sumof positive and negative reinforcement contributions that are receivedby the neuron via two spiking channels (y⁺(t) and y⁻(t)):

F _(reinf) =y ⁺(t)−y ⁻(t),  (Eqn. 64)

where the subtraction of spike trains is understood as in Eqn. 65.Reinforcement may be generated according to the task that is beingsolved by the neuron.

A composite cost function for simultaneous reinforcement and supervisedlearning may be constructed using a linear combination of contributionsprovided by Eqn. 63 and Eqn. 64:

$\begin{matrix}{F = {{{aF}_{\sup} + {bF}_{reinf}}=={{a{\int_{- \infty}^{t}{\left( {\sum\limits_{i}^{\;}\; {{\delta \left( {t - t_{i}} \right)}^{{- {({t - s})}}/\tau_{d}}\ {s}}} \right)\left( {{\sum\limits_{i}^{\;}\; {{\delta \left( {t - t_{i}^{d}} \right)}{t}}} - C} \right)}}} + {{b\left( {{\sum\limits_{j}^{\;}\; {{\delta \left( {t - t_{i}^{+}} \right)}{t}}} - {\sum\limits_{j}^{\;}{{\delta \left( {t - t_{j}^{-}} \right)}{t}}}} \right)}.}}}} & \left( {{Eqn}.\mspace{14mu} 65} \right)\end{matrix}$

Using the description of Eqn. 65, the spiking neuron network (e.g., thenetwork 700 of FIG. 7) may be configured to maximize the combined costfunction F_(sr) using one or more of the methodologies described in aco-owned and co-pending U.S. patent application entitled “APPARATUS ANDMETHODS FOR IMPLEMENTING LEARNING RULES USING PROBABILISTIC SPIKINGNEURAL NETWORKS” filed contemporaneously herewith, and incorporatedsupra.

FIGS. 9A-9F present data related to simulation results of the spikingnetwork (e.g., the network 700 of FIG. 7) configured in accordance withsupervised and reinforcement rules described with respect to Eqn. 65,supra. The input into the network (e.g., the neuron 730 of FIG. 7) isshown in the panel 900 of FIG. 9A and may comprise a single100-dimensional input spike stream of length 600 ms. The horizontal axisdenotes elapsed time in milliseconds, the vertical axis denotes eachinput dimension (e.g., the connection 714 in FIG. 7), each rowcorresponds to the respective connection, and dots denote individualspikes within each row. The panel 902 in FIG. 9A, illustrates supervisorsignal, comprising a sparse 600 ms-long stream of training spikes,delivered to the neuron 730 via the connection 704, in FIG. 7. Each dotin the panel 902 denotes the desired output spike y^(d)(t).

The reinforcement signal may be provided to the neuron according to thefollowing protocol:

-   -   If the network (e.g., the network 700 of FIG. 7) generates one        spike within a [0.50 ms] time window from the onset of the        input, then it may receive the positive reinforcement spike,        illustrated in the panel 904 in FIG. 9A.    -   If the network does not generate outputs during that interval or        generates more than one spike, then it may receive negative        reinforcement spike, illustrated in the panel 906 in FIG. 9A.    -   If the network is active (generates output spikes) during time        intervals [200 ms, 250 ms] and [400 ms, 450 ms], then it may        receive negative reinforcement spike.    -   Reinforcement signals may not be generated during one or more        other intervals.        A maximum reinforcement configuration may comprise (i) one        positive reinforcement spike and (ii) no negative reinforcement        spikes. A maximum negative reinforcement configuration may        comprise (i) no positive reinforcement spikes and (ii) three        negative reinforcement spikes.

The output activity (e.g., the spikes y(t)) of the network 660 prior tolearning, illustrated in the panel 910 of FIG. 9A, shows that output 910comprises few output spikes generated at random times that do notdisplay substantial correlation with the supervisor input 902. Thereinforcement signals 904, 906 show that the untrained neuron does notreceive positive reinforcement (manifested by the absence of spikes inthe panel 904) and receives two spikes of negative reinforcement (shownby the dots at about 50 ms and about 450 ms in the panel 906) becausethe neuron is quiet during [0 ms-50 ms] interval and it spikes during[400 ms-450 ms] interval.

FIG. 9B illustrates output activity of the network 700, operatedaccording to the supervised learning rule, which may be effected bysetting the coefficients (a,b) of Eqn. 65 as follows: a=1, b=0.Different panels in FIG. 9B present the following data: panel 900depicts feed-forward input into the network 700 of FIG. 7; panel 912depicts supervisor (training) spiking input; and panels 914, 916 depictpositive and negative reinforcement input spike patterns, respectively.

The output of the network shown in the panel 910 displays a bettercorrelation (compared to the output 910 in FIG. 9A) of the network withthe supervisor input. Data shown in FIG. 9B confirm that while thenetwork learns to repeat the supervisor spike pattern it fails toperform reinforcement task (receives 3 negative spikes—maximum possiblereinforcement).

FIG. 9C illustrates output activity of the network, operated accordingto the reinforcement learning rule, which may be effected by setting thecoefficients (a,b) of Eqn. 65 as follows: a=0, b=1. Different panels inFIG. 9C present the following data: panel 900 depicts feed-forward inputinto the network; panel 922 depicts supervisor (training) spiking input;panels 924, 926 depict positive and negative reinforcement input spikepatterns, respectively.

The output of the network, shown in the panel 920, displays no visiblecorrelation with the supervisor input, as expected. At the same time,network receives maximum possible reinforcement (one positive spike andno negative spikes) illustrated by the data in panels 924, 926 in FIG.9C.

FIG. 9D illustrates output activity of the network 700, operatedaccording to the reinforcement learning rule augmented by the supervisedlearning, effected by setting the coefficients (a,b) of Eqn. 65 asfollows: a=0.5, b=1. Different panels in FIG. 9D present the followingdata: panel 900 depicts feed-forward input into the network; panel 932depicts supervisor (training) spiking input; and panels 934, 936 depictpositive and negative reinforcement input spike patterns, respectively.

The output of the network shown in the panel 930 displays a bettercorrelation (compared to the output 910 in FIG. 9A) of the network withthe supervisor input. Data presented in FIG. 9D show that networkreceives maximum possible reinforcement (panel 934, 936) and beginsstarts to reproduce some of the supervisor spikes (at around 400 ms and470 ms) when these do not contradict with the reinforcement learningsignals. However, not all of the supervised spikes are echoed in thenetwork output 930, and additional spikes are present (e.g., the spikeat about 50 ms), compared to the supervisor input 932.

FIG. 9E illustrates data obtained for an equal weighting of supervisedand reinforcement learning: (a=1; b=1 in of Eqn. 65). The reinforcementtraces 944, 946 of FIG. 9E show that the network receives maximumreinforcement. The network output (trace 940) contains spikescorresponding to a larger portion of the supervisor input (the trace942) when compared to the data shown by the trace 930 of FIG. 9E,provided the supervisor input does not contradict the reinforcementinput. However, not all of the supervised spikes of FIG. 9E are echoedin the network output 940, and additional spikes are present (e.g., thespike at about 50 ms), compared to the supervisor input 942.

FIG. 9F illustrates output activity of the network, operated accordingto the supervised learning rule augmented by the reinforcement learning,effected by setting the coefficients (a,b) of Eqn. 65 as follows: a=1,b=0.4. The output of the network shown in the panel 950 displays abetter correlation with the supervisor input (the panel 952), ascompared to the output 940 in FIG. 9E. The network output (950) is shownto repeat the supervisor input (952) event when the latter contradictswith the reinforcement learning signals (traces 954, 956). Thereinforcement data (956) of FIG. 9F show that while the network receivemaximum possible reinforcement (trace 954), it is penalized (negativespike at 450 ms on trace 956) for generating output that is inconsistentwith the reinforcement rules.

Combined Supervised and Unsupervised Learning Tasks

In some implementations, in accordance with the framework described by,inter alia, Eqn. 43, the cost function F_(su), corresponding to acombination of supervised and unsupervised learning tasks, may beexpressed as follows:

F _(su) =aF _(sup) +c(−F _(unsup)).  (Eqn. 66)

where F_(sup) is described by, for example, Eqn. 34, F_(unsup) is thecost function for the unsupervised learning tasks, and a, c arecoefficients determining relative contribution of each cost component tothe combined cost function. By varying the coefficients a, c duringdifferent simulation runs of the spiking network, effects of relativecontribution of individual learning methods on the network learningperformance may be investigated.

In order to describe the cost function of the unsupervised learning, aninstantaneous Kullback-Leibler divergence between two point processesmay be used:

F _(unsup)=ln(p(t))−ln(p ^(d)(t))  (Eqn. 67)

where p(t) is the probability of the actual spiking pattern generated bythe network, and p^(d)(t) is the probability of a spiking patterngenerated by Poisson process. The unsupervised learning task in thisimplementation may serve to minimize the function of Eqn. 67 such thatwhen the two probabilities p(t)=p^(d)(t) are equal at all times, thenthe network may generate output spikes according to Poissondistribution.

Accordingly, the composite cost function for simultaneous unsupervisedand supervised learning may be expressed as a linear combination of Eqn.63 and Eqn. 67:

$\begin{matrix}\begin{matrix}{F = {{aF}_{\sup} + {c\left( {- F_{unsup}} \right)}}} \\{= {{a\left\lbrack {{\int_{\infty}^{t}\left( {\int_{\infty}^{t}{\left( {\sum\limits_{i}^{\;}\; {{\delta \left( {t - t_{i}} \right)}^{\frac{t - s}{\tau_{d}}}\ {s}}} \right){\sum\limits_{i}^{\;}\; {{\delta \left( {t - t_{i}^{d}} \right)}\ {t}}}}} \right)} - C} \right\rbrack} +}} \\{{c\left( {{\ln \left( {p^{d}(t)} \right)} - {\ln \left( {p(t)} \right)}} \right)}}\end{matrix} & \left( {{Eqn}.\mspace{14mu} 68} \right)\end{matrix}$

Referring now to FIGS. 8A-8C, data related to simulation results of thespiking network 700 may be configured in accordance with supervised andunsupervised rules described with respect to Eqn. 68, supra. The inputinto the neuron 730 is shown in the panel 1000 of FIG. 10A-10C and maycomprise a single 100-dimensional input spike stream of length 600 ms.The horizontal axis denotes elapsed time in ms, the vertical axisdenotes each input dimension (e.g., the connection 714 in FIG. 7), anddots denote individual spikes.

FIG. 10A illustrates output activity of the network (e.g., network 700of FIG. 7), operated according to the supervised learning rule, which iseffected by setting the coefficients (a,c) of Eqn. 68 as follows: a=1,b=0. The panel 1002 in FIG. 10A, illustrates supervisor signal,comprising a sparse 600 ms-long stream of training spikes, delivered tothe neuron 730 via the connection 704 of FIG. 7. Dot in the panel 1002denotes the desired output spike y^(d)(t).

The output activity (the spikes y(t)) of the network, illustrated in thepanel 1010 of FIG. 10A, shows that the network successfully repeats thesupervisor spike pattern which does not behave as a Poisson process with60 Hz firing rate.

FIG. 10B illustrates output of the network, where supervised learningrule is augmented by 15% fraction of Poisson spikes, effected by settingthe coefficients (a,c) of Eqn. 68 as follows: a=1, c=0.15. The outputactivity of the network, illustrated in the panel 1020 of FIG. 10B,shows that the network successfully repeats the supervisor spike pattern1022 and further comprises additional output spikes are randomlydistributed and the total number of spikes is consistent with thedesired firing rate.

FIG. 10C illustrates output of the network 700, where supervisedlearning rule is augmented by 80% fraction of Poisson spikes, effectedby setting the coefficients (a,c) of Eqn. 68 as follows: a=1, c=0.8. Theoutput activity of the network 700, illustrated in the panel 1030 ofFIG. 10B, shows that the network output is characterized by the desiredPoisson distribution and the network tries to repeat the supervisorpattern, as shown by the spikes denoted with circles in the panel 1030of FIG. 10C.

Combined Supervised, Unsupervised, and Reinforcement Learning Tasks

In some implementations, in accordance with the framework described by,inter alia, Eqn. 43, the cost function F_(sur), representing acombination of supervised, unsupervised, and/or reinforcement learningtasks, may be expressed as follows:

F _(sur) =aF _(sup) +bF _(reinf) +c(−F _(unsup))  (Eqn. 69)

Referring now to FIG. 11, data related to simulation results of thespiking network configured in accordance with supervised, reinforcement,and unsupervised rules described with respect to Eqn. 69, supra. Thenetwork learning rules comprise equally weighted supervised andreinforcement rules augmented by a 15% fraction of Poisson spikes,representing unsupervised learning. Accordingly, the weight coefficientsof Eqn. 69 are set as follows: a=1; b=1; c=0.1.

In FIG. 11, panel 1100 depicts the input comprising a single100-dimensional input spike stream of length 600 ms; panel 902 depictsthe supervisor input; and panels 904, 906 depict positive and negativereinforcement inputs into the network 700 of FIG. 7, respectively.

The network output, presented in panel 1110 in FIG. 11, comprises spikesthat generated based on (i) reinforcement learning (the first spike at50 ms leads to the positive reinforcement spike at 60 ms in the panel1104); (ii) supervised learning (e.g., spikes between 400 ms and 500 msinterval); and (iii) random activity spikes due to unsupervised learning(e.g., spikes between 100 ms and 200 ms interval).

Supervised Learning Tasks

FIG. 12 presents to simulation results of the spiking network (e.g., thenetwork 700 of FIG. 7) configured in accordance with supervised learningrule. In one implementation, the costs function, comprising a measurebetween the desired y^(d) (t) and the actual y(t) output spike train ofthe neuron, may be determined using low-pass filtering of the desiredand actual spike trains, as follows:

$\begin{matrix}{{{F\left( {x,y} \right)} = {{d\left( {y,y^{d}} \right)} = \left( {{\int_{T}^{\;}{{y(s)}{a\left( {t - s} \right)}\ {s}}} - {\int_{T}^{\;}{{y^{d}(s)}{b\left( {t - s} \right)}\ {s}}}} \right)^{2}}}\mspace{79mu} {{where},}} & \left( {{Eqn}.\mspace{14mu} 70} \right) \\{\mspace{79mu} {{{y(t)} = {\sum\limits_{i}^{\;}\; {\delta \left( {t - t_{i}^{out}} \right)}}},\mspace{79mu} {{y^{d}(t)} = {\sum\limits_{j}^{\;}\; {\delta \left( {t - t_{j}^{d}} \right)}}},}} & \left( {{Eqn}.\mspace{14mu} 71} \right)\end{matrix}$

x is the input signal, δ is the Dirac delta function, t_(i) ^(out),t_(j) ^(d) are the output and desired spike times, respectively, anda(t), b(t) are some positive finite-response kernels. In one or moreimplementations, one or both kernels (e.g., a(t)) may comprise anexponential trace:

a(t)=e ^(−t/τ) ^(a) ,  (Eqn. 72)

where τ_(a) is the time interval, typically selected according toaverage output firing rate of a neuron.

The results, depicted in FIG. 12 correspond to the input signal {X}comprised of 100 input spike trains (shown in the panel 1200 in FIG.12); the network is able to reproduce the desired spike train (shown inthe panel 1202), as illustrated by very close match of the networkoutput shown in the panel 1210.

Predictive Supervised Learning Tasks

The generalized learning framework described herein is not limited tothe applications characterized by an immediate correspondence betweenthe network activity and the cost function. In one or moreimplementations, the present framework may be applied to a spikeprediction task, using spike train distance as the cost functionconfigured as:

F(x,y)=∫_(T)(∫_(−∞) ^(t) y(s)α(t−s)ds)Σ_(j)δ(t−t _(j) ^(d))dt  (Eqn. 73)

where α(t) is the alpha function. As shown by Eqn. 73, first, the actualoutput spike train y(t) may be low-pass filtered via a convolution withthe alpha-function kernel. This may create an output trace that reachesa maximum value after a time delay of τ_(d) from the output spikeoccurrence. Subsequently, the value of the filtered trace may beevaluated at the desired spike times t_(j) ^(d). The resulting costfunction F may reach its maximum when spike output precede desiredoutput (i.e., supervisory input) by the time interval of τ_(d).Accordingly, the network learns to maximize the cost function of Eqn. 73in order to predict the desired spikes with the exact delay. Thepredictive supervised learning may be used, in some implementations, fora variety of prediction tasks such as, for example, building forwardmodels.

FIG. 13 presents simulation results of the spiking network (e.g., thenetwork 700 of FIG. 7) configured in accordance with predictivesupervised learning rule of Eqn. 73. The results depicted in FIG. 13correspond to the input signal {X} comprised of 100 input spike trains(shown in the panel 1300 in FIG. 13); the network is able to predict thedesired spike train (shown in the panel 1302), as illustrated closematch of the network output shown in the panel 1310, and indicated bythe arrows 1304 in FIG. 13. Notice that the network predictive outputy(t) is not the exact shifted replica of the desired signal patterny^(d)(t), as may be seen from comparing data in the panels 1302, 1310 inFIG. 13. This is due to, partly, the low-pass filter kernel, embodied inthe performance function F which ‘smooth’ out the presence of multipleclosely-spaced spikes, indicated by spike group marked with the arrow1306 in FIG. 13.

Reciprocal Supervised and Learning Tasks

In some implementations, supervised learning may be used to in order tocause pauses in the activity of the neuron prior to generating thedesired spikes, also referred to as the reciprocal supervised learning.This task may be formalized as minimization of the performance functionF of Eqn. 73 with a constant non-associative potentiation of thesynaptic weights (e.g., the weights 712 in FIG. 7. The non-associativepotentiation may lead to a gradual increase of the firing rate of theneuron (performing exploration), while the associative minimization ofthe function F may cause pauses with a certain delay before thesupervised spike, as illustrated in FIG. 14. The results depicted inFIG. 14 correspond to the input signal {X} comprised of 100 input spiketrains (shown in the panel 1400 in FIG. 14). The network output (shownin the panel 1402) comprises periods of inactivity (pauses) denoted bybroken lines 1406_1, 1406_2 in FIG. 14, preceding network output pulseassociated with respective desired output pulse of the training signaly^(d)(t) (the association being indicated by the arrows 1404 in FIG.14). It is noteworthy that multiple consecutive pulses in the trainingsignal y^(d)(t) (e.g., the pulse group 1408) cause longer period ofinactivity, as indicated by the inactivity period 1406_1, as compared tothe inactivity period 1406_1, that is associated with a single pulse inthe training signal y^(d)(t).

Unsupervised Learning Tasks Mutual Information Maximization

In one or more implementations, the learning framework of the disclosuremay be applied to unsupervised learning where the cost function F isconfigured based on inputs x and outputs y(t) of the network (e.g., thenetwork 700 of FIG. 7). In some implementations of unsupervisedlearning, the mutual information I(x,y) between the input x and output yspike trains of the networks may be used as the cost function F, sothat:

F

=I(x,y)=

ln(p(y|x)/p(y))

_(x,y)

F(x,y)=h(y)−h(y|x)  (Eqn. 74)

where h(y) is the unconditional per-stimulus entropy (surprisal),described by (Eqn. 13). Learning by the network 700 may be configured tomaximize the cost function F of Eqn. 74.

FIG. 15A presents simulation results of the spiking network (e.g., thenetwork 700 of FIG. 7) configured in accordance with unsupervisedlearning rule of Eqn. 74. The results depicted in FIG. 15A correspond tothe input signal {X} comprised of 100 input spike trains (shown in thepanel 1500 in FIG. 15). The network output (shown in the panel 1510)show that neurons activity does not decay to zero and does not increaseuncontrollably, so that the neuron is capable of extracting informationfrom the input 1500. The panel 1502 illustrates evolution of weightsduring learning. While some of the weights shown in the panel 1502exhibit continuing growth, the resulting activity of the network (panel1510) is not adversely affected, as the average of the weights remainsconstrained.

Minimization of Kullback-Leibler Divergence

In some implementations of unsupervised learning, the performancefunction may be configured to minimize the Kullback-Leibler divergence(D_(KL)) between the output spike train distribution p(y) and thedesired probability distribution p^(d)(y). The performance function maybe expressed as:

F

=D _(KL)(p∥p ^(d))=

ln(p(y)/p ^(d)(y))

_(x,y)

F(x,y)=h ^(d)(y)−h(y)  (Eqn. 75)

In one or more implementations, the desired output y^(d)(t) ischaracterized as the Poisson point process with average firing rate rand p^(d)(y) comprises Poisson distribution. This configuration ofresults in a constrained average firing rate of the neuron (e.g., theneuron 730 in FIG. 7), thereby preventing the weights from growinginfinitely large. Furthermore, Poisson distribution of the desiredoutput, causes output y(t) with exponential distribution of inter-spikeintervals (ISI). The Poisson process is the point process with thelargest entropy for that particular firing rate and therefore it is themost informative point process. This means that minimizing theKullback-Leibler divergence between the output distribution and thePoisson distribution causes maximization of information transmission bythe network subject to the firing rate (energy) constraint.

FIGS. 15B-15C present simulation results of the spiking network (e.g.,the network 700 of FIG. 7) configured in accordance with unsupervisedlearning rule of Eqn. 75, corresponding to the input signal {X}comprised of 100 input spike trains, (shown in the panel 1540 in FIG.15C). Each input spike train in the panel 1540 is characterized byPoisson distribution with 50-Hz average firing rate.

Network averaged performance and weight evolution with time are shown inthe panels 1530, 1532, of FIG. 15B, respectively. The performance shownin FIG. 15B corresponds to maximization of negative divergence, hencethe performance increases over time. The weight evolution in the panel1532 illustrates weight stabilization after the best (stable)performance is achieved. As shown by the neuron weight evolution (panel1532 in FIG. 15B) and network output) the KL-divergence minimizationensures that weight do not grow substantially after the desiredperformance (average filing rate) is achieves. Accordingly, firing rateof the neurons (panel 1542 in FIG. 15C) remains controlled and does notincrease infinitely with time.

Reinforcement Learning Tasks

One or more implementations of reinforcement learning may requiresolving adaptive control task (e.g., AUV/UAV navigation) without havingdetailed prior information about the dynamics of the controlled plant(e.g., the plant 514 in FIG. 5). The reinforcement signal (e.g., thesignal 504 in FIG. 5) is typically used to specify to the adaptivecontroller (e.g., the controller 520 of FIG. 5) whether prior behaviorled to “desired” or “undesired” results.

FIG. 16 illustrates operation of the neural network of FIG. 7,configured to control navigation of an autonomous unmanned vehicle (AUV)along a trajectory using reinforcement learning. Data in the top panel1600 illustrate mean distance <d> between the actual position of an AUVy(t) and the desired position of the AUV. Data in the bottom panel 1610present variance of the distance d. Data in FIG. 16 illustrate improvednetwork operation, characterized by a decrease of the position meanerror and variance with time due to learning rules that enable providesminimization of average costs (maximization of average performance).

FIGS. 17A-17B illustrate operation of the neural network of FIG. 7,configured to implement coincidence detector. The neuron 730 of FIG. 7may be configured to receive two spiking inputs, presented by the traces1700-1, 1700_2 in FIG. 17A. The output of the network and the rewardsignal are presented in the panels 1710, 1702 in FIG. 17A, respectively.The neuron 730 may be configured to generate an output spike only whenit receives two input spikes simultaneously and remain silent otherwise.When the neuron performs coincidence detection (e.g., it spikes at theright time, as illustrated for example by the arrow 1712 in FIG. 17A),it receives a positive reward spike (reinforcement equals to one,illustrated by the arrow 1704 in FIG. 17A). If a neuron does notgenerate a spike when the two input spikes are present, or if it spikesafter only one input, the reinforcement may be negative (e.g.,reinforcement equals to zero).

The learning process is depicted in FIG. 17B, where the panel 1720displays performance, determined by an average normalized reinforcementthrough epoch (epoch equals to 250 s); and the panel 1730 presentsevolution of weights during learning. Performance measure in FIG. 17B isconfigured as a ratio of correct coincidence detections by neuron and itranges from 0.5 (corresponding to random guesses: no detection), to 1.0,when each output spike is associated with coincident input (perfectdetection). As seen from the data shown in FIG. 17B, the performancegradually increases due partly to increase in weights and after about2000 s weight change stabilizes and performance remains within a rangebetween 0.75 and 0.95. The network operation illustrated in FIG. 17B, isadvantageously enabled by learning rules that provide minimization ofaverage costs (maximization of average performance).

Exemplary Uses and Applications of Certain Aspects of the Invention

Generalized learning framework apparatus and methods of the disclosuremay allow for an improved implementation of single adaptive controllerapparatus system configured to simultaneously perform a variety ofcontrol tasks (e.g., adaptive control, classification, objectrecognition, prediction, and/or clasterisation). Unlike traditionallearning approaches, the generalized learning framework of the presentdisclosure may enable adaptive controller apparatus, comprising a singlespiking neuron, to implement different learning rules, in accordancewith the particulars of the control task.

In some implementations, the network may be configured and provided toend users as a “black box”. While existing approaches may require endusers to recognize the specific learning rule that is applicable to aparticular task (e.g., adaptive control, pattern recognition) and toconfigure network learning rules accordingly, a learning framework ofthe disclosure may require users to specify the end task (e.g., adaptivecontrol). Once the task is specified within the framework of thedisclosure, the “black-box” learning apparatus of the disclosure may beconfigured to automatically set up the learning rules that match thetask, thereby alleviating the user from deriving learning rules orevaluating and selecting between different learning rules.

Even when existing learning approaches employ neural networks as thecomputational engine, each learning task is typically performed by aseparate network (or network partition) that operate task-specific(e.g., adaptive control, classification, recognition, prediction rules,etc.) set of learning rules (e.g., supervised, unsupervised,reinforcement). Unused portions of each partition (e.g., motor controlpartition of a robotic device) remain unavailable to other partitions ofthe network even when the respective functionality of not needed (e.g.,the robotic device remains stationary) that may require increasedprocessing resources (e.g., when the stationary robot is performingrecognition/classification tasks).

When learning tasks change during system operation (e.g., a roboticapparatus is stationary and attempts to classify objects), generalizedlearning framework of the disclosure may allow dynamic re-tasking ofportions of the network (e.g., the motor control partition) atperforming other tasks (e.g., visual pattern recognition, or objectclassifications tasks). Such functionality may be effected by, interalia, implementation of generalized learning rules within the networkwhich enable the adaptive controller apparatus to automatically use anew set of learning rules (e.g., supervised learning used inclassification), compared to the learning rules used with the motorcontrol task. These advantages may be traded for a reduced networkcomplexity, size and cost for the same processing capacity, or increasednetwork operational throughput for the same network size.

Generalized learning methodology described herein may enable differentparts of the same network to implement different adaptive tasks (asdescribed above with respect to FIGS. 5B-5C). The end user of theadaptive device may be enabled to partition network into differentparts, connect these parts appropriately, and assign cost functions toeach task (e.g., selecting them from predefined set of rules orimplementing a custom rule). The user may not be required to understanddetailed implementation of the adaptive system (e.g., plasticity rulesand/or neuronal dynamics) nor is he required to be able to derive theperformance function and determine its gradient for each learning task.Instead, the users may be able to operate generalized learning apparatusof the disclosure by assigning task functions and connectivity map toeach partition.

Furthermore, the learning framework described herein may enable learningimplementation that does not affect normal functionality of the signalprocessing/control system. By way of illustration, an adaptive systemconfigured in accordance with the present disclosure (e.g., the network600 of FIG. 6A or 700 of FIG. 7) may be capable of learning the desiredtask without requiring separate learning stage. In addition, learningmay be turned off and on, as appropriate, during system operationwithout requiring additional intervention into the process ofinput-output signal transformations executed by signal processing system(e.g., no need to stop the system or change signals flow.

In one or more implementations, the generalized learning apparatus ofthe disclosure may be implemented as a software library configured to beexecuted by a computerized neural network apparatus (e.g., containing adigital processor). In some implementations, the generalized learningapparatus may comprise a specialized hardware module (e.g., an embeddedprocessor or controller). In some implementations, the spiking networkapparatus may be implemented in a specialized or general purposeintegrated circuit (e.g., ASIC, FPGA, and/or PLD). Myriad otherimplementations may exist that will be recognized by those of ordinaryskill given the present disclosure.

Advantageously, the present disclosure can be used to simplify andimprove control tasks for a wide assortment of control applicationsincluding, without limitation, industrial control, adaptive signalprocessing, navigation, and robotics. Exemplary implementations of thepresent disclosure may be useful in a variety of devices includingwithout limitation prosthetic devices (such as artificial limbs),industrial control, autonomous and robotic apparatus, HVAC, and otherelectromechanical devices requiring accurate stabilization, set-pointcontrol, trajectory tracking functionality or other types of control.Examples of such robotic devices may include manufacturing robots (e.g.,automotive), military devices, and medical devices (e.g., for surgicalrobots). Examples of autonomous navigation may include rovers (e.g., forextraterrestrial, underwater, hazardous exploration environment),unmanned air vehicles, underwater vehicles, smart appliances (e.g.,ROOMBA®), and/or robotic toys. The present disclosure can advantageouslybe used in other applications of adaptive signal processing systems(comprising for example, artificial neural networks), including: machinevision, pattern detection and pattern recognition, objectclassification, signal filtering, data segmentation, data compression,data mining, optimization and scheduling, complex mapping, and/or otherapplications.

It will be recognized that while certain aspects of the disclosure aredescribed in terms of a specific sequence of steps of a method, thesedescriptions are only illustrative of the broader methods of theinvention, and may be modified as required by the particularapplication. Certain steps may be rendered unnecessary or optional undercertain circumstances. Additionally, certain steps or functionality maybe added to the disclosed implementations, or the order of performanceof two or more steps permuted. All such variations are considered to beencompassed within the disclosure disclosed and claimed herein.

While the above detailed description has shown, described, and pointedout novel features of the disclosure as applied to variousimplementations, it will be understood that various omissions,substitutions, and changes in the form and details of the device orprocess illustrated may be made by those skilled in the art withoutdeparting from the disclosure. The foregoing description is of the bestmode presently contemplated of carrying out the invention. Thisdescription is in no way meant to be limiting, but rather should betaken as illustrative of the general principles of the invention. Thescope of the disclosure should be determined with reference to theclaims.

What is claimed:
 1. Stochastic spiking neuron signal processing systemconfigured to implement task specific learning, the system comprising: acontroller apparatus configured to generate output control signal ybased at least in part on input signal, said controller apparatuscharacterized by a controller state parameter, and a control parameterand a learning apparatus configured to: generate an adjustment signalbased at least in part on said input signal, said controller stateparameter, and said output signal; and provide said adjustment signal tosaid controller apparatus, thereby effecting said learning; wherein:said control parameter is configured in accordance with said task; andsaid adjustment signal is configured to modify said control parameterbased at least in part on said input signal and said output signal. 2.The system of claim 1, wherein said output control signal comprises aspike train configured based at least in part said adjustment signal;and said learning apparatus comprises: a task-specific block, configuredindependent from the controller state parameter, the task-specific blockconfigured to implement said task-specific learning; and acontroller-specific block, configured independent from saidtask-specific learning; wherein said task-specific learning ischaracterized by a performance function, said performance functionconfigured to effect at least unsupervised learning rule.
 3. The systemof claim 2, further comprising a teaching interface operably coupled tosaid learning apparatus and configured to provide a teaching signal;wherein: said teaching signal comprises a desired controller outputsignal; and wherein said performance function is further configured toeffect a supervised learning rule, based at least in part on saiddesired controller output signal.
 4. The system of claim 3, wherein:said teaching signal further comprises a reinforcement spike trainassociated with current performance of the controller apparatus relativeto desired performance; and said performance function is furtherconfigured to effect a reinforcement learning rule, based at least inpart on said reinforcement spike train.
 5. The system of claim 4,wherein: said current performance is based at least in part onadjustment of said control parameter from a prior state to currentstate; said reinforcement is positive when said current performance iscloser to desired performance of said controller; and said reinforcementis negative when said current performance is farther from said desiredperformance.
 6. The system of claim 4, wherein said task-specificlearning comprises a hybrid learning rule comprising a combination ofsaid reinforcement, said supervised and said unsupervised learning rulessimultaneous with one another.
 7. The system of claim 1, wherein saidadjustment signal is determined as a product of controller performancefunction with a gradient of per-stimulus entropy parameter, saidgradient is determined with respect to said control parameter; andper-stimulus entropy parameter is configured to characterize dependenceof said output signal on (i) said input signal; and (ii) said controlparameter.
 8. The system of claim 7, wherein said per-stimulus entropyparameter is determined based on a natural logarithm of p(y|x,w), wherep denotes conditional probability of said output signal given said inputsignal x with respect to said control parameter w.
 9. Computer readableapparatus comprising a storage medium, said storage medium comprising aplurality of instructions to adjust a learning parameter associated witha computerized spiking neuron configured to produce output spike signalconsistent with (i) an input spike signal, and (ii) a learning task, theinstructions configured to, when executed: construct time derivativerepresentation of a trace of a neuron state, based at least in part onsaid input spike signal and a state parameter; obtain a realization ofsaid trace, based at least in part in integrating said time derivativerepresentation; and determine adjustment of said learning parameter,based at least in part on said trace; wherein said adjustment isconfigured to transition said neuron state towards a target state, saidtarget state associated with said neuron generating said output spikesignal.
 10. The apparatus of claim 9, wherein said integrating saidrepresentation is effected via symbolic integration operation.
 11. Theapparatus of claim 9, wherein: said state parameter is configured tocharacterize time evolution of said neuron state; said realization ofsaid trace comprises an analytic solution of said time derivativerepresentation; and said construct of said time derivativerepresentation enables to attain said integration via symbolicintegration operation.
 12. The apparatus of claim 9, wherein: said stateparameter q is configured to characterize time evolution of said neuronstate in accordance with a state evolution process characterized by: aresponse mode and a transition mode, said response mode is associatedwith generating a neuronal response; state transition term V describingchanges of neuronal state in said transition mode; state transition termR describing changes of state set in said response mode; and statetransition term G describing changes of state set due to said input. 13.The apparatus of claim 12, wherein: said state parameter is configuredto characterize neuron membrane voltage; and said input comprises analogsignal and said state transition term G is configured to describechanges of said voltage due to said analog signal.
 14. The apparatus ofclaim 12, wherein said input comprises a plurality of spikes and saidstate transition term G is configured to describe changes of said stateparameters due to an integral effect of said plurality of spikes; andsaid output comprises at least said neuronal response.
 15. The apparatusof claim 12, wherein: said state parameter q comprises neuronexcitability and; said time derivative representation comprises a sum ofV, R, G state transition terms each multiplicatively combined with saidtrace.
 16. The apparatus of claim 12, wherein said state transition termV comprises said trace multiplicatively combined with a Jacobian matrixJv configured in accordance with said transition mode of said evolutionprocess; state transition term R comprises said trace multiplicativelycombined with a Jacobian matrix Jr configured in accordance with saidresponse mode of said evolution process; and state transition term Gcomprises said trace multiplicatively combined with a Jacobian matrix Jgconfigured in accordance with said input x.
 17. The apparatus of claim9, wherein: said input comprises feed-forward input into said neuron viaan interface; and said learning parameter comprises efficacy associatedwith said interface.
 18. The apparatus of claim 17, wherein saidinterface comprises synaptic connection and said learning parametercomprises connection weight.
 19. The apparatus of claim 9, wherein saidstate parameter q is configured to describe time evolution of saidneuron state in accordance with a state evolution process, saidevolution process characterized by an instantaneous probability density(IPD) of generating neuronal response; said instructions are furtherconfigured to, when executed, determine derivative of said IPD, withrespect to said learning parameter, based at least in part on said traceand obtain an instantaneous score function value, based at least in parton said derivative of said IPD; and said determine said adjustment isbase at least in part on said instantaneous score function value. 20.The apparatus of claim 19, wherein: said determine said realization ofsaid trace, said determine said derivative of said IPD, and said obtaininstantaneous score function value cooperate to produce said adjustmentsuch that a next instance of said neuron state, associated with anadjusted value, configured based on said current value and saidadjustment, is closer to said target state.
 21. A computerized apparatusconfigured to process input spike train using hybrid learning rule, theapparatus comprising stochastic learning block configured to producelearning signal based at least in part on said input spike and trainingsignal; wherein said hybrid learning rule is configured tosimultaneously effect reinforcement learning rule and unsupervisedlearning rule.
 22. The apparatus of claim 21, wherein said stochasticlearning block is operable according to a stochastic processcharacterized by a current state and a desired state, said process beingdescribed by at least a state variable configured to transition saidlearning block from current state to said desired state; said trainingsignal r comprises a reinforcement spiking indicator associated withcurrent performance relative to desired performance of the apparatus,said current performance corresponding to said current state and saiddesired performance corresponding to said desired state; said currentperformance is effected, at least partly, by transition from a priorstate to said current state; said reinforcement learning is configuredbased at least in part on said reinforcement spiking indicator so thatit provides: positive reinforcement when a distance measure between saidcurrent state and said desired state is smaller compared to saiddistance measure between said prior state and said desired state; andnegative reinforcement when said distance measure between said currentstate and said desired state is greater compared to said distancemeasure between said prior state and said desired state.
 23. Theapparatus of claim 22, wherein: said training signal further comprisesdesired output spike train; current performance is effected, at leastpartly, by transition from prior state to said current state; and saidreinforcement learning is configured based at least in part on saidreinforcement spiking indicator so that: positive reinforcement whensaid current performance is closer to said desired performance; and saidreinforcement is negative when said current performance is farther fromsaid desired performance.
 24. The apparatus of claim 22, wherein: saidstochastic learning block is operable according to stochastic processcharacterized by current state and desired state, said process beingdescribed by at least state variable configured to transition saidlearning block from current state to said desired state; said hybridlearning rule is further configured to simultaneously effectreinforcement learning rule, unsupervised learning rule, and supervisedlearning rule; said hybrid learning rule is characterized by a hybridperformance function, comprising a simultaneous combination ofreinforcement learning performance function, supervised learningperformance function, and unsupervised learning performance function;and said simultaneous combination is effectuated by at least in part ona value of said hybrid performance function determined at a time step t,said value comprising reinforcement performance function value,supervised learning performance function value, and unsupervisedlearning performance function value.